On the sign of the main diagonal of the symmetric part of a matrix.

Question

New question! Hope its interesting enough.

Assume $M\in\mathbb{R}^{n\times n}$ and define the symmetric-part operator $\mathscr{S}\{\bullet\}$ such as,

$\mathscr{S}\{M\} = \dfrac{M+M^T}{2}$

I'm interested in the sign of the elements of the main diagonal of $\mathscr{S}\{M^{-1}\}$, denoted (for a moment) as $\mathit{signd}\left \{\mathscr{S}\{M^{-1}\}\right \}$. Particularly, I was wondering if there is any relation between $\mathit{signd}\left \{\mathscr{S}\{M^{-1}\}\right \}$ and $\mathit{signd}\left \{\mathscr{S}\{M\}\right \}$.

I know about the existence of a few majorization theorems concerning positive-definite or negative-definite matrices, which relates the diagonal elements of a symmetric matrix and its eigenvalues (and therefore, the sign of them). But, what happens if $\mathscr{S}\{M\}$ is neither positive-definite, nor negative-definite?

Thanks!

Answer 1

Hints:

1. It's probably a good idea to just say "I'm looking at symmetric matrices", and forget the general matrices and projections onto the symmetric subset, at least as a first step.

2. For that class -- where $M$ is already symmetric -- the SVD might be a helpful way to get started on things. You can write $M = U D U'$ (where prime denotes transpose, $U$ is orthogonal, and $D$ diagonal. But then $M^{-1} = U D^{-1} U'$, and it's pretty clear that the signs of the diagonal elements of $D$ and $D^{-1}$ are the same. This might also tell you something about the signs of the diagonal elements of $M$.

3. In general, the number of positive minus the number of negative eigenvalues is called the "signature" of $M$ (where we think of $(u, v) \mapsto u' M v$ as a quadratic form); there are various theorems about the invariance of the signature under certain kinds of transformations.

None of these is an answer to your question, of course, but I hope that they may be of some help.