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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?

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No, for example the matrix

$$A = \begin{pmatrix} 0 & 1 & 0\\ 0 & -1 & 1\\ 0 & 0 & -1 \end{pmatrix} $$ is a counterexample, since $rk(A+I) = 2$, so there's only a 1-dimensional space of eigenvectors for the eigenvalue $-1$, that has algebraic multiplicity 2.

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