# Principal submatrices of a positive definite matrix

Let $$\mathbf{A}\in\mathbb{R}^{N \times N}$$ be symmetric positive definite. For some $$1\leq k, partition $$\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?

• You can self-study the term of leading principal submatrices, really useful because it points out A11 and A22 are symmetric positive definite by definition. Good luck on you! Commented Jul 17, 2018 at 21:06

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).