I asked this question in a slightly different form and didn't receive any comments. Two real, symmetric, positive semidefinite matrices $A$ and $B$ contain the same elements (in different orders). They also have the same characteristic polynomial,
$\mathrm{det} \left[ A - \lambda I \right] = \mathrm{det} \left[ B - \lambda I \right]$.
Does this imply that the matrices are related by a signed permutation,
$A$ = $PBP^{T}$,
where $P$ is the signed permutation?
I am not a mathematician, so I have no idea if this question is trivial to people in the know. Any advice would be appreciated.
Edit:
In a comment, Ben Grossmann pointed out this related question,
Can you completely permute the elements of a matrix by applying permutation matrices?
One of the answers there says the following,
Given two elements $_1$ and $_2$, the properties "$_1$ and $_2$ are on different rows" and "$_1$ and $_2$ are on different columns" are preserved by any permutation.
Rephrasing my original question, if matrix $A$ satisfies the property above and matrix $B$ does not, do they necessarily have different characteristic polynomials?