# What Is a Permutation Matrix?

A permutation matrix is a square matrix in which every row and every column contains a single $1$ and all the other elements are zero. Such a matrix, $P$ say, is orthogonal, that is, $P^TP = PP^T = I_n$, so it is nonsingular and has determinant $\pm 1$. The total number of $n\times n$ permutation matrices is $n!$.

Premultiplying a matrix by $P$ reorders the rows and postmultiplying by $P$ reorders the columns. A permutation matrix $P$ that has the desired reordering effect is constructed by doing the same operations on the identity matrix.

Examples of permutation matrices are the identity matrix $I_n$, the reverse identity matrix $J_n$, and the shift matrix $K_n$ (also called the cyclic permutation matrix), illustrated for $n = 4$ by

$\notag I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad J_4 = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}, \qquad K_4 = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$

Pre- or postmultiplying a matrix by $J_n$ reverses the order of the rows and columns, respectively. Pre- or postmultiplying a matrix by $K_n$ shifts the rows or columns, respectively, one place forward and moves the first one to the last position—that is, it cyclically permutes the rows or columns. Note that $J_n$ is a symmetric Hankel matrix and $K_n$ is a circulant matrix.

An elementary permutation matrix $P$ differs from $I_n$ in just two rows and columns, $i$ and $j$, say. It can be written $P = I_n - (e_i-e_j)(e_i-e_j)^T$, where $e_i$ is the $i$th column of $I_n$. Such a matrix is symmetric and so satisfies $P^2 = I_n$, and it has determinant $-1$. A general permutation matrix can be written as a product of elementary permutation matrices $P = P_1P_2\dots P_k$, where $k$ is such that $\det(P) = (-1)^k$.

It is easy to show that $\det(\lambda I - K_n) = \lambda^n - 1$, which means that the eigenvalues of $K_n$ are $1, w, w^2, \dots, w^{n-1}$, where $w = \exp(2\pi\mathrm{i}/n)$ is the $n$th root of unity. The matrix $K_n$ has two diagonals of $1$s, which move up through the matrix as it is powered: $K_n^i \ne I$ for $i< n$ and $K_n^n = I$. The following animated gif superposes MATLAB spy plots of $K_{64}$, $K_{64}^2$, …, $K_{64}^{64} = I_{64}$.

The shift matrix $K_n$ plays a fundamental role in characterizing irreducible permutation matrices. Recall that a matrix $A\in\mathbb{C}^{n\times n}$ is irreducible if there does not exist a permutation matrix $P$ such that

$\notag P^TAP = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix},$

where $A_{11}$ and $A_{22}$ are square, nonempty submatrices.

Theorem 1. For a permutation matrix $P \in \mathbb{R}^{n \times n}$ the following conditions are equivalent.

• $P$ is irreducible.
• There exists a permutation matrix $Q$ such that $Q^{-1}PQ = K_n$
• The eigenvalues of $P$ are $1, w, w^2, \dots, w^{n-1}$.

One consequence of Theorem 1 is that for any irreducible permutation matrix $P$, $\mathrm{rank}(P - I) = \mathrm{rank}(K_n - I) = n-1$.

The next result shows that a reducible permutation matrix can be expressed in terms of irreducible permutation matrices.

Theorem 2. Every reducible permutation matrix is permutation similar to a direct sum of irreducible permutation matrices.

Another notable permutation matrix is the vec-permutation matrix, which relates $A\otimes B$ to $B\otimes A$, where $\otimes$ is the Kronecker product.

A permutation matrix is an example of a doubly stochastic matrix: a nonnegative matrix whose row and column sums are all equal to $1$. A classic result characterizes doubly stochastic matrices in terms of permutation matrices.

Theorem 3 (Birkhoff). A matrix is doubly stochastic if and only if it is a convex combination of permutation matrices.

In coding, memory can be saved by representing a permutation matrix $P$ as an integer vector $p$, where $p_i$ is the column index of the $1$ within the $i$th row of $P$. MATLAB functions that return permutation matrices can also return the permutation in vector form. Here is an example with the MATLAB lu function that illustrates how permuting a matrix can be done using the vector permutation representation.

>> A = gallery('frank',4), [L,U,P] = lu(A); P
A =
4     3     2     1
3     3     2     1
0     2     2     1
0     0     1     1
P =
1     0     0     0
0     0     1     0
0     0     0     1
0     1     0     0
>> P*A
ans =
4     3     2     1
0     2     2     1
0     0     1     1
3     3     2     1
>> [L,U,p] = lu(A,'vector'); p
p =
1     3     4     2
>> A(p,:)
ans =
4     3     2     1
0     2     2     1
0     0     1     1
3     3     2     1


For more on handling permutations in MATLAB see section 24.3 of MATLAB Guide.

## Notes

For proofs of Theorems 1–3 see Zhang (2011, Sec. 5.6). Theorem 3 is also proved in Horn and Johnson (2013, Thm. 8.7.2).

Permutations play a key role in the fast Fourier transform and its efficient implementation; see Van Loan (1992).