6 (10) Let $A, B$ and $D$ be real matrices. Suppose that $A = A^\top$, $D = D^\top$ and $A$ is invertible. Then show that $\left[ \begin{array}{cc} A & B \\ B^\top & D \end{array} \right]$ is positive if and only if $A$ and $D - B^\top A^{-1}B$ are positive definite.

阿摩線上測驗 · 2026-11-13T14:39:46+08:00

6 (10) Let $A, B$ and $D$ be real matrices. Suppose that A = A^→p, D = D^→p and $A$ is invertible. Then show that [ arraycc A & B \ B^→p & D array ] is positive if and only if $A$ and D - B^→p A^-1B are positive definite.

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6 (10) Let $A, B$ and $D$ be real matrices. Suppose that $A = A^\top$, $D = D^\top$ and $A$ is invertible. Then show that $\left[ \begin{array}{cc} A & B \\ B^\top & D \end{array} \right]$ is positive if and only if $A$ and $D - B^\top A^{-1}B$ are positive definite.

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