Let $A = [a_{ij}]_{n\times n}$ be a real matrix with the property $a_{ij}a_{ji} = 0$. What can be said about the eigenvalues of$A$ ? I want to know when $A$ is non-singular and when $A$ is nilpotent. (my motivation is that$A$ can be considered as a generalized strictly upper triangular matrix, which is always nilpotent).
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$\begingroup$ Many permutation matrices have that property. You might find similarity to an upper triangular matrix a more useful generalization wrt nilpotency. $\endgroup$– The Masked AvengerCommented Aug 11, 2013 at 8:38
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