Is a principal submatrix of a diagonalizable matrix diagonalizable?

Question

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!