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Function Repository Resource:

 CommutationMatrix

Source Notebook

Generate a commutation matrix

Contributed by: Jan Mangaldan

ResourceFunction["CommutationMatrix"][m,n]

returns the commutation matrix .

Details and Options

The commutation matrix is also known as the vec-permutation matrix or the swap matrix.
The commutation matrix is a special permutation matrix of size that relates the Kronecker products and . Specifically, if a is a matrix of size p×q and b is a matrix of size r×s, then the two Kronecker products satisfy the relationship .
By default, ResourceFunction["CommutationMatrix"] produces a PermutationMatrix and takes the same options as PermutationMatrix, including TargetStructure.

Examples

Basic Examples (2) 

A commutation matrix:

In[1]:=
cm = ResourceFunction["CommutationMatrix"][5, 3]
Out[1]=

Visualize the commutation matrix:

In[2]:=
ArrayPlot[cm]
Out[2]=

Options (5) 

TargetStructure (3) 

Return the commutation matrix as a dense matrix:

In[3]:=
ResourceFunction["CommutationMatrix"][2, 3, TargetStructure -> "Dense"]
Out[3]=

Return the commutation matrix as a structured array:

In[4]:=
ResourceFunction["CommutationMatrix"][2, 3, TargetStructure -> "Structured"]
Out[4]=

Return the commutation matrix as a sparse array:

In[5]:=
ResourceFunction["CommutationMatrix"][2, 3, TargetStructure -> "Sparse"]
Out[5]=

WorkingPrecision (2) 

Specify the working precision for the commutation matrix:

In[6]:=
ResourceFunction["CommutationMatrix"][2, 3, WorkingPrecision -> MachinePrecision]
Out[6]=

The normal representation returns a full matrix with finite precision elements:

In[7]:=
Normal[%]
Out[7]=

Properties and Relations (5) 

The commutation matrix is orthogonal, i.e. the inverse is equal to the transpose:

In[8]:=
p = 4; q = 3; cm = ResourceFunction["CommutationMatrix"][p, q]; Inverse[cm] === Transpose[cm]
Out[10]=

The inverse of CommutationMatrix[p,q] is given by CommutationMatrix[q,p]:

In[11]:=
p = 4; q = 3; ResourceFunction["CommutationMatrix"][p, q] . ResourceFunction["CommutationMatrix"][q, p] == IdentityMatrix[p q]
Out[12]=

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

In[13]:=
vec[m_?MatrixQ] := Flatten[m, {{2, 1}}]

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

In[14]:=
p = 4; q = 3; m1 = Array[\[FormalX], {p, q}]; vec[Transpose[m1]] == ResourceFunction["CommutationMatrix"][p, q] . vec[m1]
Out[15]=

The commutation 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:

In[16]:=
p = 4; q = 3; m1 = Array[\[FormalX], {p, q}]; r = 4; s = 3; m2 = Array[\[FormalY], {r, s}]; KroneckerProduct[m1, m2] . ResourceFunction["CommutationMatrix"][q, s] == ResourceFunction["CommutationMatrix"][p, r] . KroneckerProduct[m2, m1]
Out[17]=

The commutation matrix can be expressed as a sum of Kronecker products of an identity matrix with unit vectors:

In[18]:=
p = 4; q = 3; ResourceFunction["CommutationMatrix"][p, q] == \!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(p\)]\(KroneckerProduct[ UnitVector[p, k], IdentityMatrix[q], UnitVector[p, k]]\)\)
Out[19]=

Requirements

Wolfram Language 13.1 (June 2022) or above

Version History

  • 1.0.0 – 16 August 2023

Source Metadata

  • Citation:
    • Henderson H.V., Searle S.R., "The vec-permutation Matrix, the vec operator and Kronecker products: a review." Linear and Multilinear Algebra, vol. 9, no. 4, 271–288, 1981.
      DOI: 10.1080/03081088108817379
    • Magnus J.R., Neudecker H., "The Commutation Matrix: Some Properties and Applications." The Annals of Statististics, vol. 7, no. 2, 381–394, 1979.
      DOI: 10.1214/aos/1176344621
    • Van Loan C.F., "The ubiquitous Kronecker product." Journal of Computational and Applied Mathematics, vol. 123, nos. 1–2, 85–100, 2000.
      DOI: 10.1016/S0377-0427(00)00393-9

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