Construct a permutation matrix from a permutation vector:

Show the elements:

Get the normal representation:

Represent a permutation matrix as a structured array:

Show the elements:

Compute the determinant:

Construct a permutation matrix from a disjoint cycle representation:

Show the elements:

Get the normal representation:

Specify the dimension of the permutation matrix:

Show the elements:

Construct a permutation matrix from a TwoWayRule (an interchange permutation):

Show the elements:

Get the normal representation:

Specify the dimension of the permutation matrix:

Show the elements:

Construct a permutation matrix from an explicit matrix:

Show the elements:

Get the normal representation:

PermutationMatrix objects include properties that give information about the array:

"PermutationCycles" gives the disjoint cycle representation of the underlying permutation:

"PermutationList" gives the permutation list representation:

"WorkingPrecision" gives the precision of the matrix entries:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

Structured algorithms are typically faster:

Compute the determinant:

Compute the inverse:

Compute the eigenvalues:

When appropriate, structured algorithms return another PermutationMatrix object:

The inverse allows you to easily get the inverse permutation:

Verify the inverse relationship:

This is equivalent to the result of InversePermutation:

Generate a random 10⁴×10⁴ permutation vector:

The representation and computation are efficient for the structured array:

The normal representation is dramatically bigger (80 KB vs. 800 MB) and slower:

The inverse needs 2.4 GB of storage:

A function for computing the LU decomposition of a matrix:

Compute the decomposition:

Verify the decomposition:

Generate permutation matrices corresponding to the elements of a permutation group:

Compute a multiplication table for the group:

This is equivalent to the result of GroupMultiplicationTable:

Define the vec operator, which stacks the columns of a matrix into a single vector:

Define the vec-permutation matrix, also called the commutation matrix:

The vec-permutation matrix relates the result of applying the vec operator to a matrix and its transpose:

The vec-permutation matrix can be used to express the relationship between the Kronecker product of two given matrices and the Kronecker product of the same matrices in reverse order:

The efficiency of the fast Fourier transform (FFT) relies on being able to form a larger Fourier matrix from two smaller ones. Generate two small Fourier matrices of sizes p and q:

The Fourier matrix of size p q can be expressed as a product of four simpler matrices:

Show that the resulting matrix is equivalent to the result of FourierMatrix:

The discrete Fourier transform of a vector can be computed by successively multiplying the factors of the Fourier matrix to the vector:

The result is equivalent to applying Fourier to the vector:

Convert an identity matrix to a permutation matrix:

Show the disjoint cycle representation:

PermutationMatrix[p<->q] is equivalent to PermutationMatrix[Cycles[{{p,q}}]]:

Use SparseArray[PermutationMatrix[…]] to get a representation as a SparseArray:

Generate a random permutation list:

Premultiplication with a permutation matrix is equivalent to using Part with a permutation list:

Postmultiplication with a permutation matrix is equivalent to using Permute with a permutation list:

The determinant of a permutation matrix is equal to the signature of the corresponding permutation list:

The permanent of the product of a permutation matrix and a general matrix is equal to the permanent of the original matrix:

An upper bidiagonal matrix:

Generate a symmetric block matrix with bidiagonal blocks (a Jordan–Wielandt matrix):

Define the "perfect shuffle" permutation:

Use the perfect shuffle to transform the Jordan–Wielandt matrix into a tridiagonal matrix (a Golub–Kahan tridiagonal matrix):

Compute the singular values of a numerical upper bidiagonal matrix:

These are equivalent to the positive eigenvalues of the Golub–Kahan tridiagonal matrix: