Take the 2-minute tour ×  Prove that a $3\times3$   is positive
definite if and only if $a > 0$, $ad - b^2 > 0 $ , $\det K > 0$.
Also prove that an $n\times n$ matrix $K > 0$ is positive
definite if and only if all the upper left square $k \times k$
subdeterminants are positive for $k = 1, \ldots, n$.  For the
first part I know that in order it to be a positive definite
then the quadratic function $q(x)$ can be transformed by
completing the squares but I don't know how to prove $a > 0$,
$ad - b^2 > 0 $ , $\det K > 0$. For the second part I have no
idea how to do.  share|improve this question Second part implies
the first –  Berci Oct 8 '12 at 21:53  1 Answer 1  up vote 3
down vote accepted  HINT Note that if $A = \begin{pmatrix}A_{11}
& A_{12}\\ A_{21} & A_{22} \end{pmatrix}$, where $A_{ij} \in
\mathbb{R}^{n_i \times n_j}$ is positive definite, then choosing
$x = \begin{pmatrix}x_1 \\ 0 \end{pmatrix}$, where $x_1 \in
\mathbb{R}^{n_1 \times 1}$, we get $$x^TAx = x_1^T A_{11}
x_1$$Hence, if $A$ is positive defi