Editing Definite matrix (section)

=== Block matrices and submatrices ===
A positive <math>2n \times 2n</math> matrix may also be defined by [[block matrix|blocks]]:
<math display="block">M =  \begin{bmatrix} A & B \\ C & D \end{bmatrix}</math>

where each block is <math>n \times n,</math> By applying the positivity condition, it immediately follows that <math>A</math> and <math>D</math> are hermitian, and <math>C = B^*.</math>

We have that <math>\mathbf{z}^* M\mathbf{z} \ge 0</math> for all complex <math>\mathbf{z},</math> and in particular for <math>\mathbf{z} = [\mathbf{v}, 0]^\mathsf{T} .</math> Then
<math display="block">\begin{bmatrix} \mathbf{v}^* & 0 \end{bmatrix} \begin{bmatrix} A & B \\ B^* & D \end{bmatrix} \begin{bmatrix} \mathbf{v} \\ 0 \end{bmatrix} = \mathbf{v}^* A\mathbf{v} \ge 0.</math>

A similar argument can be applied to <math>D,</math> and thus we conclude that both <math>A</math> and <math>D</math> must be positive definite. The argument can be extended to show that any [[Matrix_(mathematics)#Submatrix|principal submatrix]] of <math>M</math> is itself positive definite.

Converse results can be proved with stronger conditions on the blocks, for instance, using the [[Schur complement#Conditions for positive definiteness and semi-definiteness|Schur complement]].