# Is a principal submatrix of a diagonalizable matrix diagonalizable?

Let $A$ be diagonalizable, i.e., $A=X \Lambda X^{-1}$ for some diagonal matrix $\Lambda$. Consider $B$ which is a principal submatrix of $A$.

1. Does there exist an invertible matrix $Y$ and a diagonal matrix $D$ such that $B=Y D Y^{-1}$ ?
2. $D$ and $\Lambda$ can be related through the interlacing property. Can $X$ and $Y$ also be related to each other. Specifically, if $X$ has small condition number, does $Y$ also have a small condition number?

Any thoughts/pointers are appreciated!

• $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$ is diagonalizable (over $\mathbb{C}$) but its upper left $2 \times 2$ submatrix is not. – Daniel Schepler Nov 1 '17 at 23:00
• @DanielSchepler Thank you for your response! What if A is diagonalizable over $\mathbb{R}$? Also what if it is diagonally dominant (but not necessarily symmetric)? – Ozzy Nov 1 '17 at 23:11