Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition
$$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T & \mathbf{A}_{22}\end{pmatrix},$$
where $\mathbf{A}_{11}$ is $k\times k$ and $\mathbf{A}_{22}$ is $(N-k)\times (N-k)$.
I'm trying to show that the principal submatrices
$\mathbf{A}_{11}$ and $\mathbf{A}_{22}$ are also symmetric positive definite.
I've been able to show that
$$\left [ \begin{array}{cc}
\mathbf{A}_{11} & \mathbf{A}_{12}\\
\mathbf{A}_{12}^T & \mathbf{A}_{22} \end{array} \right ] = \mathbf{A} = \mathbf{A}^T = \left [ \begin{array}{cc}
\mathbf{A}_{11}^T & \mathbf{A}_{12}\\
\mathbf{A}_{12}^T & \mathbf{A}_{22}^T \end{array} \right ]$$
This implies that $\mathbf{A}_{11} = \mathbf{A}_{11}^T$, and $\mathbf{A}_{22} = \mathbf{A}_{22}^T$. Thus, $\mathbf{A}_{11}$ and $\mathbf{A}_{22}$ are symmetric.
I'm struggling to show that $\mathbf{A}$ is positive definite. Does anyone have any ideas?
Best Answer
You know that positive definiteness means $v^TAv>0$ for all nonzero vectors $v$. Choose $v$ to be vectors with non-zero entries only at the first $k$ positions. (And then do the opposite).