Permutation matrix
This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices
Definition
A square matrix is termed a permutation matrix if it satisfes both these conditions:
- Every row has exactly one nonzero entry, and the value of that entry is 1.
- Every column has exactly one nonzero entry, and the value of that entry is 1.
Note that assuming that the matrix is an invertible matrix, the two conditions are equivalent (and moreover, all permutation matrices are invertible). Thus, a permutation matrix can be defined as an invertible matrix satisfying condition (1), or equivalently, as an invertible matrix satisfying condition (2).
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| monomial matrix | |FULL LIST, MORE INFO | |||
| signed permutation matrix | |FULL LIST, MORE INFO | |||
| doubly stochastic matrix | |FULL LIST, MORE INFO | |||
| row-stochastic matrix | |FULL LIST, MORE INFO | |||
| column-stochastic matrix | |FULL LIST, MORE INFO |