# If $A$ is positive definite then any principal submatrix of $A$ is positive definite

If $A$ is positive definite then any principal submatrix of $A$ is positive definite.

## Proof;

In the proof i dont understand about $j_i$'s.. Can some one interpret the proof in a simpler way(may be the original one is simpler)

if someboy is rewriting the proof it would give me a better understanding.(if possible)

• Looks like the author defined principal submatrix as one with the same indices of columns and rows removed. Could that be a problem? Nov 29 '15 at 10:36

The $j_i$'s represent the coordinates removed from $A$ to obtain the principal submatrix $B$. So we want to show $y^{\intercal}By > 0$ for all non-zero $y \in \mathbb{R}^{n-s}$. The argument shows that any such $y$ can be exported to some $x \in \mathbb{R}^n$ (by choosing the missing coordinates as zero) such that $y^{\intercal}By = x^{\intercal}Ax$. Then the result follows.