Generalized permutation matrix

In matrix theory, a generalized permutation matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column.

An example of generalized permutation matrix is

${\begin{bmatrix}0&0&3&0\\0&-2&0&0\\1&0&0&0\\0&0&0&1\end{bmatrix}}$

It is known that if a nonnegative matrix with a nonnegative inverse is a generalized permutation matrix.