Commutation matrix
Encyclopedia
In mathematics
, especially in linear algebra
and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix
into the vectorized form of its transpose
. Specifically, the commutation matrix K(m,n) is the mn × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT):
Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:
where A = [Ai,j].
The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product
: for every m × n matrix A and every r × q matrix B,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
into the vectorized form of its transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
. Specifically, the commutation matrix K(m,n) is the mn × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT):
- K(m,n) vec(A) = vec(AT) .
Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:
- vec(A) = [ A1,1, ..., Am,1, A1,2, ..., Am,2, ..., A1,n, ..., Am,n ]T
where A = [Ai,j].
The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product
Kronecker product
In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...
: for every m × n matrix A and every r × q matrix B,
- K(r,m)(A
B)K(n,q) = B 
A.