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m Fixing the fact that this refers to a positive definite matrix but uses \ge. Tag: Reverted |
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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}
<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}
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.
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