Permutation matrix

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A.

Definition[]

Given a permutation π of m elements,

${\displaystyle \pi :\lbrace 1,\ldots ,m\rbrace \to \lbrace 1,\ldots ,m\rbrace }$

represented in two-line form by

${\displaystyle {\begin{pmatrix}1&2&\cdots &m\\\pi (1)&\pi (2)&\cdots &\pi (m)\end{pmatrix}},}$

there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the m × m identity matrix, I_m, either permute the columns or permute the rows, according to π. Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to the other representation. This article will primarily deal with just one of these representations and the other will only be mentioned when there is a difference to be aware of.

The m × m permutation matrix P_π = (p_ij) obtained by permuting the columns of the identity matrix I_m, that is, for each i, p_ij = 1 if j = π(i) and p_ij = 0 otherwise, will be referred to as the column representation in this article.^[1] Since the entries in row i are all 0 except that a 1 appears in column π(i), we may write

${\displaystyle P_{\pi }={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\vdots \\\mathbf {e} _{\pi (m)}\end{bmatrix}},}$

where ${\displaystyle \mathbf {e} _{j}}$, a standard basis vector, denotes a row vector of length m with 1 in the jth position and 0 in every other position.^[2]

For example, the permutation matrix P_π corresponding to the permutation ${\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}}}$ is

${\displaystyle P_{\pi }={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\\\mathbf {e} _{4}\\\mathbf {e} _{2}\\\mathbf {e} _{5}\\\mathbf {e} _{3}\end{bmatrix}}={\begin{bmatrix}1&0&0&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&1&0&0\end{bmatrix}}.}$

Observe that the jth column of the I₅ identity matrix now appears as the π(j)th column of P_π.

The other representation, obtained by permuting the rows of the identity matrix I_m, that is, for each j, p_ij = 1 if i = π(j) and p_ij = 0 otherwise, will be referred to as the row representation.

Properties[]

The column representation of a permutation matrix is used throughout this section, except when otherwise indicated.

Multiplying ${\displaystyle P_{\pi }}$ times a column vector g will permute the rows of the vector:

${\displaystyle P_{\pi }\mathbf {g} ={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\vdots \\\mathbf {e} _{\pi (n)}\end{bmatrix}}{\begin{bmatrix}g_{1}\\g_{2}\\\vdots \\g_{n}\end{bmatrix}}={\begin{bmatrix}g_{\pi (1)}\\g_{\pi (2)}\\\vdots \\g_{\pi (n)}\end{bmatrix}}.}$

Repeated use of this result shows that if M is an appropriately sized matrix, the product, ${\displaystyle P_{\pi }M}$ is just a permutation of the rows of M. However, observing that

${\displaystyle P_{\pi }\mathbf {e} _{k}^{\mathsf {T}}=\mathbf {e} _{\pi ^{-1}(k)}^{\mathsf {T}}}$

for each k shows that the permutation of the rows is given by π⁻¹. (${\displaystyle M^{\mathsf {T}}}$ is the transpose of matrix M.)

As permutation matrices are orthogonal matrices (i.e., ${\displaystyle P_{\pi }P_{\pi }^{\mathsf {T}}=I}$), the inverse matrix exists and can be written as

${\displaystyle P_{\pi }^{-1}=P_{\pi ^{-1}}=P_{\pi }^{\mathsf {T}}.}$

Multiplying a row vector h times ${\displaystyle P_{\pi }}$ will permute the columns of the vector:

${\displaystyle \mathbf {h} P_{\pi }={\begin{bmatrix}h_{1}&h_{2}&\cdots &h_{n}\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\vdots \\\mathbf {e} _{\pi (n)}\end{bmatrix}}={\begin{bmatrix}h_{\pi ^{-1}(1)}&h_{\pi ^{-1}(2)}&\cdots &h_{\pi ^{-1}(n)}\end{bmatrix}}}$

Again, repeated application of this result shows that post-multiplying a matrix M by the permutation matrix P_π, that is, M P_π, results in permuting the columns of M. Notice also that

${\displaystyle \mathbf {e} _{k}P_{\pi }=\mathbf {e} _{\pi (k)}.}$

Given two permutations π and σ of m elements, the corresponding permutation matrices P_π and P_σ acting on column vectors are composed with

${\displaystyle P_{\sigma }P_{\pi }\,\mathbf {g} =P_{\pi \,\circ \,\sigma }\,\mathbf {g} .}$

The same matrices acting on row vectors (that is, post-multiplication) compose according to the same rule

${\displaystyle \mathbf {h} P_{\sigma }P_{\pi }=\mathbf {h} P_{\pi \,\circ \,\sigma }.}$

To be clear, the above formulas use the prefix notation for permutation composition, that is,

${\displaystyle (\pi \,\circ \,\sigma )(k)=\pi \left(\sigma (k)\right).}$

Let ${\displaystyle Q_{\pi }}$ be the permutation matrix corresponding to π in its row representation. The properties of this representation can be determined from those of the column representation since ${\displaystyle Q_{\pi }=P_{\pi }^{\mathsf {T}}=P_{{\pi }^{-1}}.}$ In particular,

${\displaystyle Q_{\pi }\mathbf {e} _{k}^{\mathsf {T}}=P_{{\pi }^{-1}}\mathbf {e} _{k}^{\mathsf {T}}=\mathbf {e} _{(\pi ^{-1})^{-1}(k)}^{\mathsf {T}}=\mathbf {e} _{\pi (k)}^{\mathsf {T}}.}$

From this it follows that

${\displaystyle Q_{\sigma }Q_{\pi }\,\mathbf {g} =Q_{\sigma \,\circ \,\pi }\,\mathbf {g} .}$

Similarly,

${\displaystyle \mathbf {h} \,Q_{\sigma }Q_{\pi }=\mathbf {h} \,Q_{\sigma \,\circ \,\pi }.}$

Matrix group[]

If (1) denotes the identity permutation, then P₍₁₎ is the identity matrix.

Let S_n denote the symmetric group, or group of permutations, on {1,2,...,n}. Since there are n! permutations, there are n! permutation matrices. By the formulas above, the n × n permutation matrices form a group under matrix multiplication with the identity matrix as the identity element.

The map S_n → A ⊂ GL(n, Z₂) is a faithful representation. Thus, |A| = n!.

Doubly stochastic matrices[]

A permutation matrix is itself a doubly stochastic matrix, but it also plays a special role in the theory of these matrices. The Birkhoff–von Neumann theorem says that every doubly stochastic real matrix is a convex combination of permutation matrices of the same order and the permutation matrices are precisely the extreme points of the set of doubly stochastic matrices. That is, the Birkhoff polytope, the set of doubly stochastic matrices, is the convex hull of the set of permutation matrices.^[3]

Linear algebraic properties[]

The trace of a permutation matrix is the number of fixed points of the permutation. If the permutation has fixed points, so it can be written in cycle form as π = (a₁)(a₂)...(a_k)σ where σ has no fixed points, then e_a1,e_a2,...,e_ak are eigenvectors of the permutation matrix.

To calculate the eigenvalues of a permutation matrix ${\displaystyle P_{\sigma }}$, write ${\displaystyle \sigma }$ as a product of cycles, say, ${\displaystyle \sigma =C_{1}C_{2}\cdots C_{t}}$. Let the corresponding lengths of these cycles be ${\displaystyle l_{1},l_{2}...l_{t}}$, and let ${\displaystyle R_{i}(1\leq i\leq t)}$ be the set of complex solutions of ${\displaystyle x^{l_{i}}=1}$. The union of all ${\displaystyle R_{i}}$s is the set of eigenvalues of the corresponding permutation matrix. The geometric multiplicity of each eigenvalue equals the number of ${\displaystyle R_{i}}$s that contain it.^[4]

From group theory we know that any permutation may be written as a product of transpositions. Therefore, any permutation matrix P factors as a product of row-interchanging elementary matrices, each having determinant −1. Thus the determinant of a permutation matrix P is just the signature of the corresponding permutation.

Examples[]

Permutation of rows and columns[]

When a matrix M is multiplied by a permutation matrix P on the left to make PM, the product is the result of permuting the rows of M. As a special case, if M is a column vector, then PM is the result of permuting the entries of M:

P · (1, 2, 3, 4)T = (4, 1, 3, 2)T

When instead M is multiplied by a permutation matrix on the right to make MP, the product is the result of permuting the columns of M. As a special case, if M is a row vector, then MP is the result of permuting the entries of M:

(1, 2, 3, 4) · P = (2, 4, 3, 1)

Permutation of rows[]

The permutation matrix P_π corresponding to the permutation ${\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}}}$ is

${\displaystyle P_{\pi }={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}={\begin{bmatrix}\mathbf {e} _{1}\\\mathbf {e} _{4}\\\mathbf {e} _{2}\\\mathbf {e} _{5}\\\mathbf {e} _{3}\end{bmatrix}}={\begin{bmatrix}1&0&0&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&1&0&0\end{bmatrix}}.}$

Given a vector g,

${\displaystyle P_{\pi }\mathbf {g} ={\begin{bmatrix}\mathbf {e} _{\pi (1)}\\\mathbf {e} _{\pi (2)}\\\mathbf {e} _{\pi (3)}\\\mathbf {e} _{\pi (4)}\\\mathbf {e} _{\pi (5)}\end{bmatrix}}{\begin{bmatrix}g_{1}\\g_{2}\\g_{3}\\g_{4}\\g_{5}\end{bmatrix}}={\begin{bmatrix}g_{1}\\g_{4}\\g_{2}\\g_{5}\\g_{3}\end{bmatrix}}.}$

Explanation[]

A permutation matrix will always be in the form

${\displaystyle {\begin{bmatrix}\mathbf {e} _{a_{1}}\\\mathbf {e} _{a_{2}}\\\vdots \\\mathbf {e} _{a_{j}}\\\end{bmatrix}}}$

where e_ai represents the ith basis vector (as a row) for R^j, and where

${\displaystyle {\begin{bmatrix}1&2&\ldots &j\\a_{1}&a_{2}&\ldots &a_{j}\end{bmatrix}}}$

is the permutation form of the permutation matrix.

Now, in performing matrix multiplication, one essentially forms the dot product of each row of the first matrix with each column of the second. In this instance, we will be forming the dot product of each row of this matrix with the vector of elements we want to permute. That is, for example, v = (g₀,...,g₅)^T,

e_ai·v = g_ai

So, the product of the permutation matrix with the vector v above, will be a vector in the form (g_a1, g_a2, ..., g_aj), and that this then is a permutation of v since we have said that the permutation form is

${\displaystyle {\begin{pmatrix}1&2&\ldots &j\\a_{1}&a_{2}&\ldots &a_{j}\end{pmatrix}}.}$

So, permutation matrices do indeed permute the order of elements in vectors multiplied with them.

Restricted forms[]

• Costas array, a permutation matrix in which the displacement vectors between the entries are all distinct
• n-queens puzzle, a permutation matrix in which there is at most one entry in each diagonal and antidiagonal

See also[]

• Alternating sign matrix
• Generalized permutation matrix
• Rook polynomial
• Permanent

References[]

• Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. 108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.
• Joseph, Najnudel; Ashkan, Nikeghbali (2010), The Distribution of Eigenvalues of Randomized Permutation Matrices, arXiv:1005.0402, Bibcode:2010arXiv1005.0402N