I am working as an undergraduate on this project where I am analyzing nxn matrices that follow these two rules:
1) Only entries that on the diagonal (i.e. the entry in row 1 column 1, row 2 column 2, etc) can be negative. All other entries must be greater than or equal to 0.
2) All columns must sum to 0
In effect, this means that all the diagonals are non-positive, since they are just the additive inverse of all the rest of the entries in the column which are all non-negative.
The interesting question has arisen: are all matrices of this form diagonalizable? I have been able to prove that every 2x2 matrix of this form is diagonalizable except the zero matrix. However, when I try to see if the pattern extends to larger matrices, things get complicated quickly.
I simply cannot find a counter-example. Does anyone knows of a counter example or any theorems/ideas for proceeding with this problem?