The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e., ( S ⊗ ⊗ T ) ( v 1 , … … , v n , v n + 1 , … … , v n + m ) = S ( v 1 , … … , v n ) T ( v n + 1 , … … , v n + m ) , which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e., ( S ⊗ ⊗ T ) j 1 … … j k j k + 1 … … j k + m i 1 … … i l i l + 1 … … i l + n = S j 1 … … j k i 1 … … i l T j k + 1 … … j k + m i l + 1 … … i l + n . If S is of type (l, k) and T is of type (n, m), then the tensor product S ⊗ T has type (l + n, k + m).