Morse index and permutation of diagonal entries of a symmetric matrix

Question

Do there exist results concerning preservation or not of the Morse index of a symmetric matrix $A$, after permuting its diagonal entries, and keeping fixed the off--diagonal ones?

Thanks!

Answer 1

You can see that the signature is not preserved by this operation considering a matrix $A$ which is a direct sum of two matrices respectively of orders 2 and 1. Say that the off-diagonal entries are all zero but $a_{12}=a_{21}=1$.For instance, it is definite positive iff $a_{11}+a_{22} > 0$ $a_{11}a_{22} > 1,$ and $a_{33} > 0$, a condition which is in general not preserved by a permutation of the diagonal.