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{{unreferenced|date=August 2009}}
In [[mathematics]], a '''generalized permutation matrix''' (or '''monomial matrix''') is a [[matrix (mathematics)|matrix]] with the same nonzero pattern as a [[permutation matrix]], i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is
:<math>\begin{bmatrix}
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0 & 0 & 0 & 1\end{bmatrix}.</math>
== Structure ==
A [[nonsingular matrix]] ''A'' is a generalized permutation matrix if and only if it can be written as a product of a nonsingular [[diagonal matrix]] ''D'' and a [[permutation matrix]] ''P'':
:<math> A=DP. </math>
== Properties ==
An interesting theorem states the following: If a nonsingular matrix and its inverse are both nonnegative matrices (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix.
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