Oh God, she got caught. She got caught. Sorry, I got Shields and Heals. Four days. Four days. What's up everybody on TheMangus?  She's ethereal's one cleric, the cleric's being the support class of the game. I'll be playing the full game this time, no fast forwarding, and this will also serve as a bit of a guide to playing Marina. I think I'm a little better than average player with her. That's not something you're going to hear me say often about too many heroes or myths as they call them. If you have any interest in trying ethereal out, there's going to be an open weekend for the game from January 21st to the 23rd. It's free and no need for a key. You just make an account of their website during that weekend, download the game and play it. I'm also probably in the future going to do a quick run down of how to play the game because there was a lot of confusion last time. There's a lot of new mechanics that are very different from Paragon so you're going to have to get used to them. But for now, let's get into some Marina gameplay. Go to Fire Lane, but I've got to see where Dante's going. It looks like we're going ice because Talos went fire, and I think Talos is going to be our off laner. Dante's already using his right click there. So yeah, we'll come over here to ice. It's pretty common to come to ice as duo lane. It's becoming more and more common for duo to go to Fire Lane though. But yeah, we'll just play this out in ice. It would be nice if we were just up against their solo laner. We could probably just dominate them because the game is designed. The myths aren't designed to go 2v1 like that or 1v2 as I should say. Which I miss from Paragon. I really loved going 1v2 in Paragon. A lot of people complained about that. I understand why they changed it and all that, but man, just hitting was at level 5 is when you got your ultimate in Paragon. At level 5 and then starting to crush people in the off lane, like crush the duo when you're 1v2 was just amazing. I loved doing it. So I unlocked Riptide first. Riptide, it's a slow. You lay a big pool of water on the ground, and it slows people, deals damage. It's really good for poke, and it's really good for defense. So I kind of like to use it, grab that early. And yeah, she just used hers on us. The thing, um... See, there we go. I don't know where her Dante is. Maybe he got confused and went to the wrong lane. Good for us. Oh, she just charmed her on under the tower. That was a good play on her part. I don't have shield yet. So that's kind of Marina's main thing. She can pop these bubble shields on people, and if you hit, that's her E, and if you hit alt E, you can put a shield on yourself. So she's going to have to stand in there and take that damage. What I do with Marina, if we're ahead, then I'll be very aggressive with her and use my abilities to poke more than for defense. But if we're behind, okay, there's their Dante. If we get behind, then I start using her abilities a little more defensively. Her Q, it spawns like this bubble. You'll see it pop out eventually. But it spawns out like this bubble that, um... It deals... there it is. It deals damage and, uh... It deals damage to enemies, and it heals allies depending on where you put it. The thing I don't like about it is you have to stop whenever you cast it. It kind of roots you in place, which isn't good in a team fight, especially when you're trying to heal somebody. But, um... Yeah, that ability, I'll poke with it a lot if we're ahead. But we're behind right now. My Dante has taken a lot of damage.  She charmed me first. So I need to use this Q to get our Dante healed back up. It's just hard because you stop, and that's the thing. You have your risk reward ability, like that charm thing that we just did. That's only a right click, and it roots you in place for quite a long time. But it forces someone, it kind of stuns them and forces them to walk toward you. So it serves as a bit of a displacement as well. But, um... You know, the risk is that you have to stand in place. But with our Q, you also have to stand in place. So it's a bit of a risk reward as well. I don't think... I don't think it's fair. I don't think it's fair. Marina's a little overpowered right now, so I don't know if that should be something they fix immediately. But I think eventually in the future, it would just feel a lot better if you could move while casting your Q. Man, I'm, uh... kind of outboxing their Dante a little bit. What's up, Marina? You want some of this too? Uh, one of the problems... Another problem I've seen from Marina is her, um, basic attacks seem to, like, come from her feet, like, the way she swings her trident. So it comes from down and goes up.  It needs to come from higher, the point of origin. So I'll just kind of bully these guys a little bit and just try and keep them back, because they're being very aggressive now. Dante's pretty much healed up, so we can... we can be a little aggressive ourselves. That was a nice charm from her. That would have been nice if I got him under... Oh, he might die, he might die. Hey! I didn't notice Malaya was there. Nice! So we got a... got us a kill. Thanks to where, uh, Malaya rotated over. Okay, don't be taking tower shots for no reason now. Don't just boost that. So she's pushed under her tower now, so I'll start using... we're both fairly topped off, so... I can start using abilities to poke.  I don't know what Talos is doing back there, but... Okay, Dante. Okay, watch her... watch her charm. Watch her charm. He's taking tower shots. Taking them tower shots. I know it's, um... like, a lot of Dante's like, dive in once they see that damage, but I could do enough poke with Riptide that they'll either, you know, back out of lane, or... it'll just kill them.  You know what I'm saying? So, yeah, he's, uh... he's pretty aggressive. Um, he's pretty... pretty good, though.    Alright, and get him shielded up.  The good thing is that I can heal him back up. Looks like, uh, Malaya wants to go in. She's gonna get that kill. Let's shield her up. She did not get the kill. Now, Dante's... Jesus. And there are Dante's here. Oh, this is, uh... this is not good. This is no bueno, so... Malaya is down. Okay, who's he fighting? Okay, he's on there. He's on there, Marina. Let's try and keep him alive while keeping ourselves alive. Stop getting... stop going under the tower, my dick! And, alright. I think I actually got that kill, but... I was trying to heal us and damage her, and I think I'd... Uh, you know, it worked, but he was just taking too many tower shots and taking shots from Dante from behind. So down we go! I like this... I like this guy's aggression, but a little... a little less tower shots would be nice. Alright, so level up our mask, um... You know what the mask does? It just regens, mana. And then, as you level it up, it gets more effects, like the Paladin's mask makes a... an area around you that... everybody takes less damage while they're in. It's not huge, but it's a... it's a pretty good radius. And then the, uh, battle spell that I'm using, uh... shaves a second off of all my cooldowns, which... I kinda like reinforcing on Marina better, just to give her an... yet another shield. But this is pretty good too.  They just placed a ward. Okay, calm down, everybody. Calm down. We'll put a ward up here, in case they, uh... try and... a lot of people try to gank from that angle. Okay, Dante, don't worry, buddy. I'm here. I'm here.  She just... used that inefficiently. Okay, he's going, he's going. Let's shield him up so... he can properly trade here, which kind of sucks, because it was Marina versus Marina, so... the shields will... shields will kinda cancel each other out. She'd be very aggressive. Let's... pull her in. Wasn't long enough.  I should probably mention her passive too, whichever... whenever she hits... an enemy with an ability, it puts a mark on their head.  It looks like a little sliver of ice over top of their head. Uh... Okay, yeah, that's probably going to cool down right now. Okay, it spawned on there, Marina. Nice kill, Dante. Nice kill. But whenever... somebody attacks that target, it gives them a burst of movement speed and healing. Um, it's... the healing is... cut in half if Marina pops the passive right.  It's kind of passably, as a passive would suggest. And, uh... Marina's cleric class ability for now is just another shield, like it's a pretty large, substantial shield. But, um... in the future, you'll be able to... pick between different, um...  Hit by their Dante's snipe. But you'll be able to pick different paths, so I guess you'll be able to do different stuff with it. I need to get myself healed up. I'll slow their assault. Oh, I hit Dante going right in after... damn, he did a lot of damage quick. He did a lot of damage quick. That's what I'm talking about.  So there's the, um... there's the cleric shield. It's a little more visible. Or less visible, I don't know. I guess it depends. It depends on how you look at it. It's bigger. It has a bigger effect. I'll slow these guys down. And, uh... that was our ultimate. You can see Dante and I are both getting healed, and anybody that's in the area as well.  Get away from me, dude. Get away from me, minion. See, that's what I mean by you have to stop to cast your Q. That minion, damn, you're cut off with me. Okay, so nice. He killed their Dante in the meantime. Well, I was running for my life from minions. Oh, he's going in right after the marina. Look at this guy go! Look at him go! All right, and so she has to either take the slow or get killed, or she's like, oh, it's a murk.  The shields block a lot of damage, but they can't just soak multiple tower shots through her shields. I was able to save her. She didn't even need to go in there, too. Goddamn! He just blasted me again.  And you can see in a theory of the towers, as you hit them, as their health goes down, they kind of melt a little bit. They actually take real damage. Look at him go! Okay, don't take the tower shots. The nice thing about the shield when you have an aggressive Dante like this is that it eats a lot of the damage from minions, which is what really gets people a lot of the time. All right, he just needs to calm down and let my pull deal some damage. He's gonna... He missed it. It was a good attempt. That thing's hard to hit. I'm blaming him. That should probably... No, don't go in there! Goddamn it. I tried to shield him. He got the kill, which is nice. But yeah, my reflection pull would have finished him off there. There was no need for him to die for that. He could have kept farming lane Is it so bad? Is it really so bad for your support to get a kill? Come on, guys! That's the Ice Wyvern. We need to take that pretty soon, whenever we can. We're gonna need one of our junglers to rotate. Maybe Malaya will come back when they're hanging out lane here. I would think that they would have been back by now. Of course, the two Dantes did kind of die at the same time, but where'd Marina go? All right, Dante. Box with him just a little bit. Here's your shield. Here's your slows going down a little bit. Here's your little heal. Little damage. Little heal. Okay, don't take the tower shots. Oh, is he ulting? Ulting back, ulting back! Oh, he did! He did ulting back! There you go! There you go! It's a lot easier to hit somebody when they're standing there than just sitting there, you know? That was funny. They traded ultimates. I don't know why he was even trying to ult. The enemy Dante was trying to ult there because we're both like really high health unless he was just trying to clear the wave, I guess. Which I guess would be smart if the tower was about to go down and all that. Just keep jamming this lane down their throat.  Let's... I swirren. I swirren. We can go take I swirren now. There needs to be an in-game chat in the game, so I can let this dude know through more than just basic attacking in the direction of where I want him to go. That we need to... I mean, that was a perfect opportunity to take the I swirren, but oh well. I mean, just... We're sitting pretty anyway. But we could be sitting even prettier. But yeah, this is a pretty good example of being an aggressive marina. If I'm... I've probably said malias there, marina many times in this video. I'm guessing. I'm betting. But yeah, this is a good example of being an aggressive marina, but you don't always play this aggressive.  Like, doing something like this, you would definitely want to save that for whenever he's getting charged to one so that he can escape instead of just using to poke somebody into the tower.  You got the cannon? You got the base cannon? Nope. Riptide. And that should finish him off. And it does. And that's how it's done. That's how you do it. No need to tower dive and stuff. Just let your support handle it. Oh, I'm gonna have to look for that cough one when I edit this. Oh, hey, what's up, Noxus? Where's your little baby girl? Oh, we got our marina over here too. Or our Malia? Goddamn it. Those two names are too similar. Valium effort. Man, this guy loves using that base cannon. Alright, let's get her. That's another problem right now too. Whenever you use... Goddamn it.  No! No! Whenever you use marina's ultimate, it kills everybody's frame rates. Oh, you're going in hard, huh? Why wasn't that punished? Okay, now I'm going to get punished. Help me! Help me, Tom Cruise! Help me, baby Jesus! Help me open room for him. Alright, let's get the bug out of here. Well, let's hope we don't get base cannon. He can't see me, so...  Oh! Oh-ho-ho-ho-ho! Was that hard Talos or was that their Talos? I don't even know. Oh, that was scary. He can popped in right out of nowhere. They need to work on the mini-map, because it's hard to tell a lot of times where stuff is. Alright, so we're going to pick up Ring of Void, which gives you damage over time on all of your abilities, which I normally would not take on marina this early, but we are doing quite a bit of bullying, so why not poke even harder if we're going to be poking a lot, you know? You know what I'm saying? We'll get the Cotton Spore Guard, which is kind of a... It depends what kind of damage you're taking, what kind of armor it provides. So it's a nice option if you don't know if you need to get magic or physical armor. Hey, we got the Ice Weaver. That was probably our Talos trying to let me know that he wanted to take Ice Weaver. I don't think I really need to even come in here. You guys got this. Dante, you can't go that way. Malaya can, you can. It's getting hard. Oh, and the base cannon takes her down. Oh, she might get that Noxus too. Look at her go. Well, she popped her passive. That should be a kill. That should be a kill. Nicely done. I think I took her kill, though. That's no bueno. I'm not going to complain about it, but I probably should have let her have that. Trigger had it sometimes. Out of here. Let's drop everything on him. Down he goes. Very nice. We're cleaning up this match. That's the third tower, so now we just have an inhibitor behind the portal here. Next to their elder.  Now, these inhibitors are, I forget what they call them. I don't think they call them inhibitors, but they deal damage spread evenly among all the targets that are in range of it, so the more minions and heroes that you have in range of that suffer, the less damage it's going to do to you. The more of those you go, yep, come here. That should be a kill. That should be a kill. Okay, that's okay. Alright, everybody's really low. Okay, they're backing out. I should back out too. Let's see if I can keep her alive.  Oh, she had her passive. Nothing's back there? Oh, God. Let's see if we can heal him up. Nice. That's actually quite a bit of healing with the ultimate.  I couldn't keep him alive. Could not keep him alive. No, I'm probably going to die.  I'm getting the tick marker, so they were definitely trying to run through my riptide. As far as I know, there's no blink in the game yet. Why are they chasing me? Can I hear skifters? Well, that is jump animation. It makes me think of somebody that tried to mount a horse, jump up from behind and mount a horse, but then the horse took off before they could land. I'll always think of that every time I see your jump. Look for it. Look for it in the video. I'm not used to talking this much while recording. Streamers are always hydrating. Nothing to do with their health, and throats are just dry. I was going to back one point in time with I don't know if I can aid for it. We can just keep pushing this lane into their inhib. I know there's another name for that thing. I just can't think of it right now. Now that we're starting to get in a little bit of trouble, we should probably pick up the Buck of Pandora, which gives me some more healing and stuff. The size of them are dependent on magical powers, so we might need to up our shields. Probably just up our armor, though, so we can survive. Oh, did they get it? They got Attrapos. That's no good. Good play on their part. But now we're pushing in their ice. Pushing in their ice lane. They did something with the mini-map. They did something with the mini-map. Right now you can... If you hit M, you can pull up the entire map, but just the regular mini-map on the side only displays the lane you're in. Makes it hard to know where everybody is. Marina, what you doing, girl? Okay, there is their Dante. We have Fire, Piper, and Buff. Okay, you win this trade. Keep your shield up. Nice. I took the kill, though.  I'm trying to kill. Oh, God. Noxus. Noxus. He'll do this to me. He should be able to take these guys.  There we go, there we go. Oh, God damn. That was bonkers. Oh, Marina got him. I don't know why he didn't back out. This guy does not back down. He never backs out, never surrenders. Okay. What's up, Marina? Hot Marina on Marina Action. Is that what you want? I don't think you have regular boys, so you should probably not do this. Hey, Noxus, come here and help me.  Alright, she's out of here now. Jesus Christ. Oh, oh, we're in. A little bit of trouble, a little bit of trouble. Especially me. Okay, there's her passive.  Shoot her up. Just keep stacking the shield as much as we can. Alright, I'm out. I'm out of shields for a little bit. Nice. Nice job, Noxus. I can't believe I kept her alive in that. Is she going in too? I gotta get her healed up. There we go. There's 202 health for you. You backing out now? You backing me out? Nah, she's still going. I'm backing out.  I have 5,000 gold?  Alright, so we'll get the Paladin's Mask. I think the Paladin's Mask will do better if you had a more mobile support. Like you could dash to people and then lay down the Paladin's Mask damage debuff. Oh, looks like they're Malaya. DC, that's unfortunate. Unfortunate. Man, our Dante is killing it. At the beginning, he was definitely testing out abilities like he'd never used them before, but yeah, he learned quick. Dante's laying it down.  There's some action going on in the middle here. Ice Lane's pushing just fine. Let's go join. Go join in the fun. Let's mount our horse that rode away without us. Where are you guys going? Those. Pretty much brought them all. Yeah, Marina's super strong, man. She's super strong. I mean, that was a 3v1, but we're going to push this tower. Come on, minions. I wish there was an item that allowed you to teleport minions like a flank or something. I hate in mobas when you're like the tower's right there.  There you go. Watch out. Whoa! Close up of Noxus boobies. That was pretty hard for her. I don't know where that giant shield that Marina had was coming from, but it looks like it's gone now. Poke at him a bit. Our Dante is over there in the Ice Lane still laying it down. They noticed him. They noticed him. We keep shoving this. He's going to provide the distraction.  The one that roops. This is going to take a while. Noxus and Marina trying to take the tower down. That's a basic attack. I mean, our Dante still did good. He pushed in on that and distracted him from this. We should be able to take it. It's not making sacrifice in vain. Hey! Look at us go. I keep thinking that they're Noxus.  Oh god.    He got me.   It's a little too strong. If you're already ahead of an ADC and then you get a stun that just occurs naturally when you're basic attacking yeah, it's no bueno. A little too strong. I see like a slow or something but not just a straight up outright stun. Get back in it here. Alright, so they're almost like they're Talos ADC as well but we also had an ADC. Connecting or rage quitting or what? I mean it's, you never know with an early alpha like this. Like you can tell sometimes when somebody rage quit because they're just like they're getting their teeth kicked in and then all of a sudden they quit. But I mean with an alpha maybe they didn't just quit maybe they actually did DC. Hard to tell, hard to tell. It's also hard to tell with the way they have things set up like you killed the receiving death. Things show up. I got you baby girl, don't worry. Don't worry. Alright, let's slow them down. Okay, so that was their Talos so maybe our Talos DC? Which would mean that our Malaya DC like a long time ago. Come to think of it, I haven't seen her since that first gank attempt. I don't know, I don't know. We may have been doing a 4v5 this entire time. Okay, okay, settle down girls. Settle down. Is he still trying to push that shit? Oh god, she got caught, she got caught. Sorry, I got shields and healers. Four days. Four days. I think their Noxus is a little better than our Noxus but our Noxus has a me. And their Noxus does not have any backing out. Like that thing says an ally has been slain but it'll like you never know who it was. I'm assuming that's our Dante because that dude's been a beast all game. Kind of why I've abandoned him. He's fine on his own.  It'll be nice when they change the map up so that Ice Lane actually looks icy and stuff. Okay, Dante, go for it. Feel the tides. I'm here to help this time.  There you go, get in there, get in there. Oh, there's their Noxus. Watch out now, watch out now. Don't get AoE down.  Oh, ho, ho, ho, ho, that was a great ultimate for their Noxus. Ha, ha, ha. God jeez, not jeez. Their Noxus is freaking good. Hey, there's our Malia. Oh, they're passive.  Oh, Malia's still going. Oh, okay, okay. Let's get this Noxus out of here.  Shit! Their Noxus is good. Malia, come back. She got him. Nice. There's Noxus. Come over here to the town. Okay, okay, okay. Now you're starting to hurt. Now you're starting to hurt. Don't care for that. Don't like it. Look at her hoping somebody comes over to help. I don't remember where Malia went.  She could have come back and killed this Noxus pretty easy.  Shit, she got me. She got me. That Noxus is really good. Really good. I think we still got this, though. We're starting to fall apart a little bit, but I think we still got this. Yeah, if you're clearing minion waves, you need to use your Cidral on the ground first with them, and then use your Lightbound Portal. Just the Cidral by itself is really good. Hold on. There you go. Don't like that, but, you know, wait until you get all the minions in it before you do that. Everybody's new to the game. They're still learning all the different little tips and tricks. All right, let's get back in here. Let's finish this off. Let's finish this game up. Mids should be pushing pretty hard, too, still, or void, I should say. Wait for me, guys. I'd hate to see them charge that portal and get decimated by the entire enemy team waiting on the other side. Yeah, they're going through. Like, I wish I could at least see... Okay, they are engaged in some people. They are engaged in battle. So we took out that thingy, dude. Inhibitor thingy. Oh, man. Oh, she landed a good one there. She landed a good one. Oh, old Dante. Oh, God.  We can get her. We can get her. We can get her. Come on. She's going to laser me through the wall. I know it. Do something, Noxus. Do something.  Oh, trees. Get out of here, trees. Ain't nobody likes you, trees. Whew. Jesus. Now, we've boned that up. Okay, Malia's taking a tropos. We should be able to end with this. Should be able to end with it. Oh, we just got an ice weapon, too, Noxus. Got some shields. Got some shields. All right, boom. There we go. Let's get some super minions and push them down their tropes. Hopefully, somebody... No, no. Okay, there she comes. There she comes. As you see, as I get close to the minions now, the way the atropos buff, they get a bit more bigger. They have more armor, deal more damage. All good things. All good things with the atropos buff. Come on. Come on. Hey, where's mine?  There she is. Malia to the rescue. Or to get killed. Look at that base cannon. See, I can heal us both up. You just be a little more confident.  Don't be confident right now. Be less confident. Okay, okay, okay. She's going in. She's going in. Okay, I can't... Okay, that's the problem with these...  With the minions being more bigger. I'm sure they're after me now. God, we just... How can we... How are we not close to this gate? Well, we're not grouping up to finish it off for one thing. Other... Malia, get out! She must use her ultimate. Okay, Naxx is back up now. Let's regroup. Now our Dante is going down. Oh, we need to re-garp. We need to re-garp. If we lose this, I'll be so upset. You leave me alone, Naxxus. Not too bad damage. Okay, we're going to lose this tower. That's okay, we've got more. We've got more towers. I might be able to get them low enough where the team's able to clean up. They took a few tower shots there. Maria's just going to heal them back up. I kind of regret going with... Ring of Void now. Probably should have gone a little more defensive. I didn't know the game was going to end up like this. Not like this, guys. Not like this. Hey, what's up, Talos? Okay, do not jump in there. Thank you. Okay, now we've got our Dante with us. We can just... okay. Malia's in the back line over there. Oof, oof. God dammit. Why does that slow? Why does that make me... Why does that make me stop? Okay. Okay, okay, okay. All right, let's try to keep him up. We can kill him. Here, you have all the things. Have all the things to help you. Base cannon took out Talos, but we took out the Maria, and we took out their Dante. Now, this is the dangerous one right here. We'll put everything on her. We'll put all the GG on her. Nice! Nice!  That triggers me just a little bit to see her not get all the minions with the... Alright, there's one down. This one came back up, unfortunately. It should go down pretty easy, though. And the more inhibs you have down the more damage the elder takes. So it is worth it to get this thing down. Where did Dante go? Dante, where are you at? All right, got it. He died? What the hell's going on here? The elder's so low. There's Dante. Hey! That was a little frustrating towards the end, but... we ended up pulling out the victory. Mangooos!


last year only 18 games three home runs 13 RBI's it just feels like the clock is ticking on him He's got to get healthy. He's got to get back out in left field and DH and he has to be productive The microscope is going to be on John Carlos Dan. John said it very well Where is the guy who played 158 games? Where is the guy who showed a lot of durability? Where is the guy who carried the Yankees when a player like Aaron judge was injured the Yankees didn't have that player last year? Injuries happen. They're part of the game Stan will be the first one to tell you that this is a proud guy Who I think wants to get back out there and remind everyone? Hey, by the way, I wanted MVP at one point in my career I'm one of the most prolific power hitters in Major League Baseball. I didn't get you chance to show that last season I need to show it in 2020, but it's on stent John is right the the fans have soured on this guy I don't always understand that I think that he is still can be a very Important part of what the Yankees are trying to accomplish this guy was 40 and 100 in his first year as a Yankee and that was disappointing to some Yeah, and I think back to 2018 and how we handled the media I was praising him because it was a very quiet and kind of direct approach with the media Last year very quiet because it wasn't on the field and I'm wondering how he comes in the spring training He's probably gonna have that chip on his shoulder that Jack is talking about like I don't need to talk about this I need to stay healthy and I need to go out there and show you if he does that obviously the Yankee fans We'll welcome them back with open arms on the


And then welcome back to theCUBE's coverage. We're back in the press room now. We were at the lake house earlier, Databricks Data Plus AI Summit. It's been a great event. Packed house 12,000 people on site, 75,000 online watching. Databricks really kind of take the next level of evolution with the industry going to beyond data. Data and AI is a story, data plus AI. It's enabling not only a new infrastructure, data infrastructure models, creating new products, new ways to think about how to create value and applications. Data, because the next guest is Jeff Denworth, who's the co-founder and CMO of Vast Data, who does a lot of work with storage and how data is thought about and is in the middle of rethinking that with your company as you guys are progressively making new things happen with future of data. Thanks for coming on theCUBE. Thanks, it's been a long time. I think you were in theCUBE, 2003.  Yeah, 2003, 2013. 2013, I'm sorry, I'm really dating myself. 2013. I don't even know who was president then, I guess it might've been the- Yeah, it definitely wasn't Trump because I remember that day very vividly. I was in Seattle for a CNCF event. But, you know, one of the things that's interesting, because you and I have talked about this before in the past, during that wave of big data, remember Hadoop, the Hadoop wave, and what made Databricks really successful was Spark made it easier and cloud was emerging and that created a whole generation of value. And we saw the same old storage, you store the data, it's in the cloud, you've got on-premise, but now it seems like a whole nother kind of replatforming of the infrastructure's happening, new younger generation talents coming in with AI. People are rethinking the future of their data and they're thinking more about platforms. What's your view on that? You guys are doing a lot of work here, thinking about how storage converts to a data platform, what's your thoughts? Yeah, so Vass is a company that was rooted in building in next generation distributed systems architecture. And in the early days, we studied organizations like Facebook and Google and it was less around big data in the way that you and I used to speak about it and more around all the real unstructured data that informed some of these really rich models, right? Data that goes well beyond the numbers and tables that you put in like a classic data warehouse, right? And so as we think about that, what we see is that at the most foundational level there are new solutions needed to help people take all these massive GPU systems that are being built and actually apply them in anger to large data reservoirs that are filled with this data that doesn't have any structure or schema in the ways that business systems would expect. And so just a few weeks ago, we announced that we were the first enterprise distributed storage systems company to be supported for NVIDIA's super pod architecture. So super pods, if you don't know, are like the biggest and the baddest of the NVIDIA machines that are being built. Those are the ones that are being deployed by the people that are basically building the most popular foundational models that you have in the market today. And that requires a very specialized type of infrastructure that up until now has been kind of the domain of the large hyperscalers. And we think there's a huge opportunity to just go democratize that to the masses. Yeah, and I think I love that democratization message but we're hearing more and more of it. We heard on stage, Mosaic ML has a lot of GPUs I heard. Databricks bought those guys for 1.3 billion. So you're seeing this bolt-on GPUs but now you got cloud scale enablement too. So you got the bolt-on with GPU support. You got on-premise hybrid cloud booming. Open source is booming. All that's kind of coming together but the data is the key to AI without the right data inputs. And that came clear in the keynote here and it's kind of an epiphany here at Databricks and I heard the same thing at MongoDB event with developers and we've heard the same with others, even Google, Snowflake and even Amazon all say the same thing. Data input drives the value and there's no property issues, privacy. This is the important part of the data. Can you explain how you see the role of data in the AI equation? Sure, so we're dealing predominantly with global organizations. And there's this conversation that is in the market around data gravity but there's also new terms that are coming up and new concepts that are coming up around things like federated learning. And essentially if I'm building some sort of healthcare patient data AI system, I may not be able to pull that data out of the different sovereign data centers that it might be in but if you can just pull the weights out and create some sort of accumulated learning from all the different sites that you've got your data strewn across then interesting things can happen. So you mentioned cloud earlier, I think about cloud more like distributed computing model more than like this guy has to go to this CSP or this CSP and our realization is that everything is strewn everywhere and there's very little interconnectedness between all of these different systems both from a processing perspective and a data perspective. And so a data platform first has to be able to kind of stitch all this together into one unified cohesive view and then allow you to go and run models that are completely distributed across the world. What's some of the infrastructure requirements to run these modern deep learning workloads because that's clearly coming. You mentioned that like what's the minimum requirement? What's the ideal infrastructure to run these modern deep learning workloads? So it's very clear that you need some sort of really strong horsepower with respect to AI processors. Most commonly that's NVIDIA, there's a few other players in the space but the prescriptions that these organizations are making are for essentially supercomputers. And so you can either consume these in the cloud as a service or you can go and deploy them on premises but you first need some big machinery and then big machinery gets coupled with big data. When I say big data I don't mean like a 50 terabyte Hadoop data set that you're sitting on but I'm talking about, we're working with customers, we have one project right now, the customer the other week forecasted 10 exabytes of data that they're going to process on. So you know with respect to the data paradigm, if you have machines that can actually process at this level of intensity, our observation, I'd be curious to hear what you think, John, is like people are now starting to architect strategies to collect tons more data because for the first time ever they can actually go and process it. I was just talking to a venture capitalist who just came in earlier and big time investor successful in the field and all those data hoarders are now looking good right now because hoarding the data and storing the data which by the way it gets cheaper and cheaper are looking good because now a lot of the LLMs and these foundation models can take advantage of that data like they've never seen before and that came out loud and clear in the keynote today where you see the data, historical data, not just real time data, historical data is now just as important as real time data because you now can mix and match with it more effectively, this is developing on data. This is a trend that's actually not hard to do now and democratizing that is the killer feature and we're seeing that play out today at this event. Yeah, I think about it as, so essentially all AI, machine learning, deep learning is just applied statistics, right? Statistical models get better as you add more and more data to the training set assuming it's good enough quality but the streaming component of this is more of a reflection of how you and I think, right? We're not like scheduling something at eight o'clock tonight to say go think, right? Go interact with the world and maybe make some realizations after that. You and I are talking, languages, our API, we're trading information in real time and you're correlating that against things you already know. That's the way that computers should work, right? You should have data that just flows in and then the system understands how to process that in real time. That's why I wanted you to come on theCUBE because that's exactly the point I think you have great insight into. Ali, Godsea talked about it today in the keynote. They're breaking down the notion of data warehouse, they're killing this war on formats, they're unifying things but at the end of the day you still got to store stuff, right?  And so what you're referring to is you got to kind of bring it together and make it fluid and more interactive. Whatever you want to call it, it's still got it, it's different, but it all starts with the data is. It's stored somewhere. So either data is landed somewhere and it's stored or it has to move somewhere, okay? So you come from a storage perspective, what needs to change? The future of AI is data, data is stored. So it's a storage solution kind of on steroids but it's not a storage solution, it's not a storage platform, it's a data platform that happens to have storage. Or am I thinking it the wrong way? What's your perspective? This is, these silos going to be gone, it's not just a storage company or an app company, it's going to be a data company. What is that future? Because storage is important, you guys know a lot about that. Well I think customers are looking for something more integrated. Storage systems are dumb, applications have no memory. And so if you think about it, there's kind of two elements. One is there's a huge division in the market right now between products and services that were designed for the era of big data and what is needed to make that next step into the era where we can actually process natural data. Data that doesn't fit in a classic data warehouse, right? And that's where the last mile of robotics lives, that's where the last mile of automation is and probably the biggest benefit to humanity. But there you need support for unstructured data, you need support for thousands of GPUs that are built on top of parallel computers that stretch all the way around the world, unifying the semantic layer with the unstructured layer. And so if you have that, then you kind of think about it and you say, okay well who's in this space? And we see there's very few full stack data platform companies that are out there, there's nobody that's addressing the deep and deep learning. And the players that are out there, some come from the engine level and they start to kind of build around that, some come from the data warehouse level and they start to build around that. We came at it from the storage perspective first, thinking and understanding that applied statistical dynamic that I talked about earlier. I said okay well if you can solve for access at any level of scale to this rich set of data that people haven't really figured out a great way to process across the enterprise, if you can solve for that, then you just build on top of that capability to go better and better. Yeah and I think that's an interesting point because the number one thing that comes up when I talk about large language models and foundation models and AI is, who do I work with? Which companies do I work with on that? And then where do I run it? And then the third question, how much does it cost? And does it perform well? So at the end of the day we're still going to have the performance challenges. Sure. Databricks has a verification kind of like feature with their GPUs, it's been authorized, it runs faster, they're just trying to see more of this SLA, policies, quality of service. Performance is huge factor in all this. It is, it is and you know, if you think about putting together stacks of a bunch of different technologies and expecting it all to work together, our thesis around APIs and integrated services is that it steps the combined solution down to the lowest common denominator and that's why we started from the bottom and we've been working our way up because performance is critical but it's not just like performance at the efficiency level, it's also in terms of scale in ways that I think people don't fully appreciate and so we have a kind of a unique perspective into the future because we're working with those hyperscalers that are building the largest of the foundation models and if we can figure out how to get those things to scale to five, 10,000 GPUs, hundreds of petabytes of infrastructure, then the hope and the expectation is we can go apply that out to the rest of the enterprise that's lagging those companies by a few years. So you guys are a storage company, storage firm that's now a platform. You have to be a platform to survive. There's a few things that we haven't yet announced about what we offer and that'll be coming. Okay, all right, so what can we expect next year? Come soon, what's the... About a month from now, right? So August 1st, we're basically going to unveil what we've been working on for seven years. It's kind of like a fly, if you know the Amazon flywheel strategy where they get something on the ground, they start making money off of it and then they keep innovating. We've had a core product architecture that we've been thinking about. Internally we call it a thinking machine, which isn't our name we came up with, but we love it. And now the components are all coming into place where we can show the world what we were really thinking about so we're super excited about that. Well, right now the market is hungry for solutions that could get the most out of the data. I love the data plus AI, democratization, you mentioned that, that's right on point. I think that the enterprises have unique needs, right? And the developers are going to have theirs. And right now, if you look at the audience of developers, the demographics shifted to a lot younger on the analytics side, they're a little bit older. You got this kind of collision between, I call the old school analytics data people and the young generation was like, clear all that out of my way, I just want to take care of it for me. And the rise of applications, just seeing data applications emerging and they got to run somewhere, they got to be stored somewhere. So looking forward to hearing more. Yeah, we think data needs to be a lot smarter than it is today. You talk a little about the data developer and we see just a huge opportunity and a great future for that concept applied. You're co-founder, you've also been in the business for a while, so you know the enterprise needs.  They're super excited, so much opportunity. And a ton of white space too, as these platforms are emerging, there's no one platform that's going to rule the world. You're going to have a combination of platforms that's winner take most. And there's so much white space, there's so many opportunities to build apps. What advice do you have for them out there watching? Well, I think the thing that we think about is not focusing on the problems that have been solved for today. There's a lot of sex appeal in the marketplace right now and the easy thing is to go chase the shiny object. But the realization that we had, even before we started the company, is that the shiny object has largely been solved by people that are far more resources than you do. So you have to look a little bit farther in the future.   And I think one of the things that I like about this mosaic act is that the team is strong, they have great chops and training, which is really hard to do, hard to get a hand on GPUs. And you got to have a good team. You got to have a great fast moving team. Well, I think overpaying is relative, right? We encounter a lot of customers that are looking to do things like apply LLMs for, you know, let's say like chat bots or things like this. And there definitely is a notion of good enough. And so as, you know, stuff gets pushed in the open source, it kind of gets pushed out into the market. And people realize, okay, well, if I'm not trying to have my, you know, my financial document scanning app also write a website for me. Maybe this kind of, this model that was trained with like a hundred million parameters is good enough. But the question becomes, you know, what do we think about, and what do we talk about five years from now? And is it large language models? I'm not sure that's the... I think it's going to be abstracted away. I think it's going to be abstracted away. I think it's going to be abstracted away. I think it's going to be abstracted away. I think it's data. I mean, at the end of the day, operational data should be invisible. Apps should be running data. Data infrastructure should be scaling. I mean, I mean... Well, we'll be talking about data for sure. But... It's in your name. We'll have to change it to whatever's next.  But think about it, five years from now, you'll have, you know, compute horsepower and data sets that are five, 10 times, 20 times more than what we have today. We're looking forward to a world where you actually have machines that can start to reason and understand the fundamentals of what they're talking about. And that's exciting. Well, I'm looking forward to what you guys are doing. I like what I'm hearing. I've heard a little bit about platforms. We'll get more in August. Thanks for coming on theCUBE. Jeff Tenworth here on theCUBE. Breaking down the future of data stored, but it's going to be intelligent. It's going to be reasoned on. It's going to be applied into applications. theCUBE bringing you all the data here. We're in the press room now. We're in the lake house earlier. We're day one of Databricks' Data Plus AI Summit. I'm John Furrier with Rob Stretching, bringing you all the action. We'll be right back. Our next guest after this short break.


For me, I think it's especially in parts of the world like this, where it's really not like a mature sort of baseline for everyone to play to. I think you have to get really comfortable making a market for yourself. And by that I mean you have to almost use language as a design tool. So don't obsess over what is user experience, what is service design, what's the difference, how do I know what the difference is? Don't worry about putting silos and boundaries around things. There is endless opportunities to do service design where Hong Kong is a great example. We don't make anything here, we don't mind anything here. Literally every single dollar that circulates in the economy is coming from services. So anything you're doing even remotely close to this space is going to involve some aspect of service design. Whether or not you can get someone to understand that service design is a thing and that it needs to be done is almost beside the point. You can tailor the language that you're using, talk in the language about what they value and then just do the design anyway and you can still create a really good opportunity for yourself.


Okay, so we begin the last time we were looking at this theorem which said that if n is a positive integer and lambda i, i going from 1 to n and lambda hat i, i going from 1 to n plus 1 are two sequences of real numbers such that these two sequences interlaced with each other, meaning that lambda hat 1 is the smallest of these numbers followed by lambda 1 followed by lambda hat 2 followed by lambda 2 etc. Up to at the very end you will have lambda n minus 1 is less than or equal to lambda hat, lambda hat n is less than or equal to lambda n, this is less than or equal to lambda hat n plus 1. So, lambda hat n plus 1 is the biggest, lambda hat 1 is the smallest and all the other numbers are in between and if we denote the diagonal matrix containing lambda 1 to lambda n as its diagonal entries as this capital lambda then there exists a real scalar, real valued scalar and an n length real valued vector y such that these other numbers lambda hat 1, lambda hat 2 up to lambda hat n plus 1 are the eigenvalues of the real symmetric matrix a hat which is lambda y, y transpose a which is an n plus 1 cross n plus 1 matrix. Okay, so we were going over the proof of this, we had gone most of the way but some part that needed to be completed. So, we will begin by filling in the rest of this proof. So, first of all lambda 1 through just to recall where we were, lambda 1 through lambda n are of course the eigenvalues of lambda and further by noting that the trace of a hat must be the summation of lambda hat i while the trace of lambda it must be equal to lambda 1 plus etcetera lambda n, we know that this value a here in order for lambda hat 1 through lambda hat n to be the eigenvalues of this matrix a must be such that a is equal to summation i equal to 1 to n plus 1 lambda hat i minus the summation i equal to 1 to n lambda i. Okay, this is always going to be greater than or equal to 0. Now, if we look at the characteristic polynomial of a hat by definition it is the determinant of ti minus a hat and if we substitute that and expand it out we were able to show that it can be written in the following form where it has a factor with t minus a minus yi squared over this t minus lambda i is showing up in the denominator here. Okay, times the product of these terms i equal to 1 to n t minus lambda i and note that this is exactly the characteristic polynomial of this matrix lambda. Now, what we want is that this p a hat of t must end up being equal to the product i equal to 1 plus 1 to n plus 1 t minus lambda hat i that will ensure that lambda hat i i going from 1 to n plus 1 are the zeros of p a hat of p and so that this matrix we can then be assured that this matrix has lambda hat 1 through lambda hat n plus 1 as its eigenvalues. Now, to do that we consider two functions the first is f of t which is the product this is our desired characteristic function which is the product i equal to 1 to n plus 1 t minus lambda hat i and g of t is the characteristic function of capital lambda and that is just the product i equal to 1 to n t minus lambda i. Now, this is an n plus 1 degree polynomial. This is an n degree polynomial. So, we can always write f of t to be some g of t times t minus some constant c that is because the t power n plus 1 term has a coefficient of 1 and the t power n has a coefficient of 1 here and so it can be written as g of t times t minus c plus some r of t. This is a remainder polynomial and this has a degree at most n minus 1. Now, if so obviously the t power n plus 1 coefficients here match already by construction but if we compare the coefficients at t to the n we did this the last time we end up with saying that c must be equal to a. Okay, now further because g of lambda k is equal to 0 for k going from 1 to n because there's a t minus lambda i factor here. What we have is f of lambda k must be equal to r of lambda k. So, basically what this means is this r of t is known to us at n points because we know what f of t is it's just this polynomial. So, we can substitute lambda k, k going from 1 to n and we can calculate the value of f of lambda k that tell us what the value of r of lambda k is and it's a degree at most n minus 1 polynomial and it's known now at one k at n different points. Now, if we assume that lambda k's are distinct and I as I mentioned we'll at the end talk about what happens when there are repeated lambda k's but if they are distinct we can write the following Lagrange interpolation formula. So, basically knowing the value r of lambda k at k equal to 1 to n we can directly write out what r of t is. r of t is this polynomial here and notice that it is it has a g of t divided by t minus lambda i and g of t itself is this product of all such factors. So, each factor will basically cancel one of these factors here and so each of these ratios is a degree n minus 1 polynomial and that is getting weighted by f of lambda i divided by g dash of lambda i and then add it together. So, overall this has a degree at most n minus 1 but this is the expression for r of t. So, basically if you find r of lambda k, k going from 1 to n, you will find that it is equal to f of lambda k. So, it meets this constraint that we have set on these on r of t. Now, if we use this formula here f of t equals g of t into t minus c plus r of t and we divide throughout by g of t then we get that f of t divided by g of t is equal to t minus a, c is equal to a minus this whole thing divided by g of t and the g of t, g of t will cancel and so you are left with okay. So, here it was plus r of t. So, I write it as minus of minus minus i equal to 1 to n minus f of lambda i divided by g dash of lambda i times 1 over t minus lambda i. Now, this thing is, so if I write it this way f of t is equal to this whole coefficient times g of t and g of t is this product here and so now if you look back at p a hat of t that also has the same form it is t minus a minus something times the product of this thing which is exactly equal to my g of t. So, f of t is now equal to the same is not f of t is now in the same form as this p a hat of t and so for these two polynomials to completely match we just need to choose y i squared to be equal to this coefficient here minus f of lambda i over g dash of lambda i. So, since f of lambda k hat equals 0 for i equal to 1 to n plus 1 we must have that if I substitute lambda k hat here this must be equal to 0 for i k equal to 1 to n plus 1. This is another condition that this polynomial ratio here will satisfy and as I said this two are of the same form. So, if you want these two to match all we need is to set y i squared to be equal to this coefficient minus f of lambda i over g dash of lambda i. So, for you to be able to find a real valued vector y such that y i squared equal to the negative of this we need that these numbers should be positive numbers. Then I can find a real valued y i such that y i squared equals this thing. So, we just need to show that these are all greater than or equal to 0 for i equal to 1 to n. So, we now finally use the interlacing assumption. Notice that in the proof so far we have not used this fact that this lambdas and lambda hats are an interlacing set of numbers. So, we will use this interlacing assumption and now if I take the so you can see here that lambda 1 is between is lower bounded by lambda hat 1 and upper bounded by lambda hat 2 and so on. So, more generally lambda i is lower bounded by lambda hat i and upper bounded by lambda hat i plus 1. So, what this means is that if I take any lambda hat j where j is less than i then lambda i minus lambda hat j will be a positive number this minus something which is on this side. And similarly if I take any j which is greater than i then lambda i minus lambda hat j is going to be negative. This is a smaller number these are all the lambda hats on this side are bigger numbers. And so, based on that if I look at what f of lambda i is the product of lambda i minus lambda j hat j going from 1 to n plus 1. But out of these all these numbers are positive and all these numbers are negative. So, there are n minus i plus 1 negative numbers over here. And so, if I pull that out I can write this as minus 1 power n minus i plus 1 product j going from 1 to n plus 1 the modulus of lambda i minus lambda j hat. And similarly g dash of lambda i is just the same product. So, g dash is this t minus lambda j but not including the ith term because you are taking the derivative and evaluating it at lambda i. And so, this product and again all these terms will be positive and all these terms will be negative. So, there are n minus i such negative terms. And so, I can write it as minus 1 to the n minus i times the product of all positive numbers mod of lambda i minus lambda j. And so, you see that this is multiplied by minus 1 to the n minus i plus 1. This is multiplied by minus 1 to the n minus i times a positive number. And so, what that means is that f of lambda i and g dash of lambda i will always have opposite signs because there is a plus 1 extra here. And so, the ratio will be negative or negative of that ratio will be positive. So, that basically establishes that we can choose y i squared to be minus f of lambda i over g dash of lambda i. So, the only thing left is to handle the case where there are repeated eigenvalues. So, this is a very simple argument. Suppose, for example, that lambda 1 equals lambda 2 which is strictly less than lambda 3 and so on. For all other cases, the argument is very similar. So, we can consider an example like this and see what happens. Now, if that is the case, because it has this interlacing property, if lambda 1 equals lambda 2, then lambda 2 hat must be equal to lambda 1 and which is in turn equal to lambda 2. So, that means that f of t, which is the product of t minus lambda hat i, it has a factor t minus lambda 2 hat, which is the same as t minus lambda 1. Similarly, g of t has two factors which are equal to t minus lambda 1. So, it has a factor t minus lambda 1 square, because the first term and the second term are both t minus lambda 1. So, the first term and the second term are t minus lambda 1 into t minus lambda 2. But since lambda 1 equals lambda 2, you have a factor like t minus lambda 1 square. And so, the multiplicity of lambda 1 as a 0 of g of t is exactly equal to 2. And so, what we can do is to modify our f of t, g of t and r of t by dividing throughout by t minus lambda 1. And this modified polynomial will be exactly like before. It will have t minus lambda 1 as a simple root of g of t. And that was the reason why we assumed that the eigenvalues are distinct, so that the eigenvalues turn out to be simple roots. And then we can reuse all of the above arguments. So, that basically is the proof. So, the proof directly extends to the case where there are repeated eigenvalues. So, this argument, the main point of this argument here is that g of t always has one greater degree for repeated roots compared to f of t. And that basically means that we can remove these common factors and then we will be left with distinct eigenvalues. And we can reuse the same arguments as we used to establish the result. Any questions about this proof? So, what did this result tell us? It told us that if you are given a set of interlacing numbers, we can construct matrices such that the first set of numbers are the eigenvalues of a matrix. And the second set of numbers are the eigenvalues of a matrix that is obtained by taking the first matrix and bordering it on the right and below by y and y transpose respectively and on the bottom right by some number a. So, and the result prior to that actually said that if you take a matrix a and then you border it by a vector y and on the right and y Hermitian below and small a on the bottom right, you will get an n plus 1 cross n plus 1 matrix. And the eigenvalues of the n cross n matrix and the n plus 1 cross n plus 1 matrix interlaced with each other. Well, there is nothing very special about adding a row and column to a matrix. The same result applies to deleting a row or a column or in fact the last row or column of a matrix. When you delete the last row or column, the eigenvalues of the reduced matrix will interlace with the eigenvalues of the original matrix. And of course, if you think about it, there is no sanctity in deleting the last row or column, nothing special about the last row or column. The results apply equally well if you delete any row or column. So, when you delete any row or column, you, so if you take an n cross n matrix and delete a particular row and column, you get what is called an n minus 1 cross n minus 1 principle sub matrix of that matrix. And the eigenvalues of a principle sub matrix interlaced with the eigenvalues of the original matrix. And you can apply this repeatedly. And then you will get interlacing results for that you will get results related to the interlacing of eigenvalues when you delete say r rows or say k rows and columns of a matrix. And so this kind of results are actually called inclusion principles. And here is one such example result. So, this is theorem. You can actually prove this theorem by using this result on adding a row and column and converting it to deleting a row and column and applying it repeatedly. But then it is equally easy to prove this result directly, which is what we will do. So, the theorem says that a is an n cross n Hermitian matrix and let r be an integer such that 1 is less than or equal to r is less than or equal to n. Let a r denote any r cross r principle sub matrix of a.  So, you have to delete the same row and column index. So, then for each integer k 1 less than or equal to k less than or equal to r lambda k of a is less than or equal to lambda k of a r is less than or equal to lambda k plus n minus r of a. Again, we are arranging these eigenvalues in increasing order. So, I didn't write that explicitly, but lambda 1 of a is the smallest eigenvalue of a and lambda n of a is the largest eigenvalue of a and lambda k of a r is the kth largest eigenvalue of the principle sub matrix a r. Okay, so proof. So, the essential ideas of this proof have we have already seen. So, we can actually quickly run through the proof. So, suppose a r is formed by deleting rows. You have to delete n minus r rows. So, i 1 up to i n minus r and the corresponding columns and let 1 less than or equal to k less than or equal to r. Okay. Now, just using the Courant-Fischer theorem, we have that. So, first of all, we want to show a lower bound like this. So, lower bound as you remember, as you might remember, we will use the min max formulation of the result. So, lambda k plus n minus r of a is equal to the min over w 1 through w n minus this index and n minus this index is n minus k minus n plus r, which is r minus k. These are in c to the n. The maximum over x not equal to 0, x in c to the n and x perpendicular to all these vectors of x Hermitian ax divided by x Hermitian x. Now, this trick is something we saw earlier. This is greater than or equal to the minimum over the same vectors w 1 through w r minus k in c to the n. The maximum over x not equal to 0, x in c to the n, x perpendicular to w 1 through w r minus k. Now, I will throw in an extra set of constraints x perpendicular to e i 1 up to e i n minus k. So, these are columns of the n cross n identity matrix corresponding to columns i 1 through i n minus k. So, I am adding extra constraints here. So, this maximum solution to this maximization problem may not be as big as the solution to this maximization problem and that is why we have a greater than or equal to sign here. So, the objective function is the same x Hermitian ax over x Hermitian x. Now, if x is perpendicular to all these vectors, then what I can do is to simply delete that corresponding row and column of a and then consider a reduced vector and then solve this over that reduced space. So, that reduced vector I will call it y and y will be perpendicular to these vectors w 1 through w n minus k, but with the indices i 1 through i n minus k deleted.  So, this is exactly equal to the minimum over vectors v 1 through v r minus k in c to the r, the maximum over y not equal to 0, y in c to the r and y perpendicular to v 1 through v r minus k of y Hermitian a r y over y Hermitian y which is exactly by the Quran Fischer theorem itself equal to lambda k of a r. So, that proves this part of the inequality and similarly lambda k of a. Now, we want to show an upper bound. So, we will use the maximine version. So, this is the maximum over w 1 through w k minus 1 in c to the n of the minimum over x not equal to 0 x in c to the n x perpendicular to w 1 through w k minus 1 of x Hermitian a x over x Hermitian x. And now I use the same trick again that this is less than or equal to if I throw in extra constraints on the minimum, I may not be able to minimize it as well as I am doing it here. So, the final answer may turn out to be larger than whatever I get here. So, this is the max over w 1 through w k minus 1 in c to the n of the minimum x not equal to 0 x in c to the n x perpendicular to w 1 through w k minus 1. And now x also perpendicular to e i 1 through e i n minus r of x Hermitian a x over x Hermitian x. But if the x is perpendicular to all these the entries i 1 through i n minus r of x are always equal to 0. So, those particular rows and columns from a can just be deleted because there is nothing to optimize over there.  So, this is exactly equal to the maximum over v 1 through v k minus 1 in c to the r the minimum over y not equal to 0 y belonging to c to the r and y perpendicular to v 1 through v k minus 1 of y Hermitian a r y over y Hermitian y which is exactly equal to lambda k of a r which is the result we wanted to show. So, that completes this proof. So, one immediate consequence of this this result is something called the Poincare separation theorem. And this is very useful for example in quantum mechanics in situations where one has information about u i Hermitian a u j for orthonormal vectors u i and u j. So, you do not get to observe the whole matrix a but you get to observe projections of this matrix using a set of orthonormal vectors. And then what can we say about the eigenvalues of the original matrix in terms of the eigenvalues of the matrix obtained by forming these projections. So, here is the corollary a is in c to the n cross n Hermitian and 1 less than or equal to r less than or equal to n. So, r is some number between 1 and n. Let u 1 through u r in c to the n be r given orthonormal vectors. And define the matrix Br with i j th element equal to u i Hermitian a u j. And notice that Br is a matrix in c to the r cross r. If the eigenvalues of a and Br are arranged in increasing order lambda k of a is less than or equal to lambda k of Br is less than or equal to lambda k plus n minus r of a. So, basically what this says is that instead of observing a, if you get to observe u i Hermitian a u j, then you arrange that in a matrix Br and then you find the. So, this is a smaller matrix possibly a much smaller matrix of size r cross r. Then you can say something about the eigenvalues of a. So, specifically the k th eigenvalue of a is at most the k th eigenvalue of Br and the n k plus n minus r th eigenvalue of a is at least equal to the k th eigenvalue of Br. So, this allows you to bound the eigenvalues of a in terms of lambda k of Br. So, the proof is very short it is just basically pointing to the previous results that we are going to use. So, if r is strictly less than n. So, if r equals n that is almost trivial, but if r is less than n I mean the next step that I am going to say is not required if r equals n because u 1 to u u n if r equals n u 1 to u n r form an n cross n orthonormal matrix together when you stack them into a matrix. But if r is less than n we can choose n minus r additional vectors u r plus 1 up to u n such that the set u 1 through u n form an orthonormal set and let u be defined as this matrix u 1 through u n in c to the n cross n. Now, u is unitary.  So, that is it. So, now we can use the previous theorem is just deleting n minus r rows and columns and we can use the previous theorem and that is exactly what the previous theorem said that the eigenvalues have this relationship. Okay. So, in fact, this result itself is very useful. You can use it to show many more results on many more variational results on eigenvalues and there are lots of results in the text. We just don't have the time to systematically cover them in this course but you can look at the text and find many more interesting results. Now, recall that the diagonal elements of Hermitian symmetric matrix are always real because if it is Hermitian symmetric a equals a Hermitian and if you equate the diagonal elements it means a must be a i must be equal to a i i star. In other words, the diagonal entries must be real value. Also, the eigenvalues of a Hermitian symmetric matrix are real value. You have seen that also. Furthermore, the diagonal entries and the eigenvalues have the same sum that is the trace of a matrix is equal to the sum of its eigenvalues.