I have been working on this problem: "Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal matrix is congruent to the original one."
I'm hoping someone can just point out a good direction to go in with it. Here's what I have tried (and failed at) so far: I have tried sandwiching a $2 \times 2$ matrix between a matrix and its transpose and worked out the product to try to find a matrix that could be generalized to the $n \times n$ case.
I also briefly hoped that any diagonal matrix and one obtained by permuting its elements would both be congruent to a matrix with $1$'s, $-1$'s, and $0$'s along the diagonal (described by my book as a canonical form for the real symmetric matrices), and congruence would then follow by congruency forming an equivalence relation and then transitivity, but this only applies (as I just said) to real symmetric matrices and does not hold for all fields.
I have also tried treating the matrix as a bilinear form and then finding a basis for which the matrix representation is a permuted version of the matrix, in which case the two matrices are congruent by the appropriate change-of-basis matrix. I got this to work out, but it involves division by the original diagonal entry and thus falls apart when that entry is zero.
I haven't found a solution yet, and not for lack of trying... Can someone point me in the right direction?