Are the following kind of matrices categorized or studied in any way? For example:
0 0 a 0 0
0 0 0 b 0
0 0 0 0 c
d 0 0 0 0
0 e 0 0 0
If diagonal matrices only scale each eigenvector, these matrices transform one eigenvector to another and then scale them.
This is a generalized permutation matrix:
https://en.m.wikipedia.org/wiki/Generalized_permutation_matrix
I know them as the normalizer of the diagonal invertible matrices, but Wikipedia has a lot more it seems.