In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. Permutation matrices are the matrix representation of permutations. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... (Redirected from (0,1) matrix) A binary matrix or (0,1)-matrix is a matrix whose entries are all either zero or one. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ...

Definition

Given a permutation π of m elements

the permutation matrix P_π with m elements is defined as

with e_i being the i-th vector in the identity matrix In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

Rules

Given two permutations π and σ of m elements and the corresponding permutation matrices P_π and P_σ

As permutation matrices are orthogonal matrices the inverse matrix exists and can be written as In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ...

The multiplication of a permutation matrix P_π with a vector g will permute the entries of the vector.

Notes

P₍₁₎ is the identity matrix. This is clear if one views the permutation matrix of a permutation σ, as the permutation of the rows or columns of the identity matrix. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

A permutation matrix is a stochastic matrix; in fact doubly stochastic. One can show that every doubly stochastic matrix is a convex linear combination of permutation matrices of the same size, giving permutation matrices a characterisation as the set of extreme points. In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a stochastic matrix is a square matrix whose columns are probability vectors, i. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In mathematics, an extreme point of a subset S of a real vector space is a point in S which does not lie in the open line segment joining any two points of S. Intuitively, an extreme point is a corner of S. Compare: concave, convex. ...

The product of a matrix M with a permutation matrix P on the left (MP) permutes the rows of M, likewise, on the right (PM), permutes the columns of M.

Since the map S_n → A ⊂ GL(n, Z₂) is a faithful representation, we have the following: Wikipedia does not yet have an article with this exact name. ... In mathematics, a group representation is a way of viewing a group in some more concrete way. ...

• There are n! n-by-n permutation matrices, so |S_n| = |A| = n!.
• The n-by-n permutation matrices form a group under matrix multiplication with the identity matrix as the identity element, as S_n is.

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 as (a₁)(a₂)...(a_n)q where q has no fixed points, then e₁,e₂,...,e_n are eigenvectors of the permutation matrix. In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Examples

The permutation matrix P_π corresponding to the permutation π=(1)(4 2 5 3) is

and given a vector g

Solving for P

The question of "if I have two adjacency matrices A & B, how can I find P?" is answerable through the use of eigenvalue decomposition which yields: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

A = QΛQ ^{− 1}

where Λ is a diagonal matrix of eigenvalues and Q is the matrix of eigenvectors. In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

The permutation matrix P relates A and B by

B = PAP ^{− 1}

Since the eigenvalues of A and B are the same then we can write:

thus Q_B = PQ_A. From this we see that the elements of an eigenvector are transformed through P.

Example

Given the two matrices

and the transformation matrix that changes A into B is

which says that the first & second row as well as the first & second column of A have been swapped to yield B (and visual inspection confirms this).

After finding the eigenvalues of both A and B and diagonalizing them into a diagonal matrix is In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...

and the Q_A matrix of eigenvectors for A is

and the Q_B matrix of eigenvectors for B is

Comparing the first eigenvector (i.e., the first column) of both we can write the first column of P by noting that the first element (Q_A(1,1) = − 0.60130) matches the second element (Q_B(2,1)), thusly we put a 1 in the second element of the first column of P. Repeating this procedure, we match the second element (Q_A(2,1)) to the first element (Q_B(1,1)), thusly we put a 1 in the first element of the second column of P; and the third element (Q_A(3,1)) to the third element (Q_B(3,1)), thusly we put a 1 in the third element of the third column of P.

The resulting P matrix is:

And comparing to the P matrix from above, we find they are the same.

Explanation

A permutation matrix will always be in the form

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

is the permutation form of the permutation matrix. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ...

Now, in performing matrix multiplication, one essentially forms the dot product of each row of the first matrix with each each column of the second. In this instance, we will be forming the dot product of each column of this matrix with the vector with elements we want to permute. That is, for example, if we call this vector 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

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

Generalization

The sum of the values in each column or row in a permutation matrix adds up to exactly 1. A possible generalization of permutation matrices are matrices where the values of each column and row add up to a number c.

For example in the following matrix M each column or row adds up to 5.

A matrix of this sort can be decomposed into permutation matrices as

with

See also

• Alternating sign matrix
• Generalized permutation matrix