Commutation matrix (original) (raw)
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 nm × 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: In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator
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| dbo:abstract | 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 nm × 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: where A = [Ai,j]. In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order. In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator (en) |
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| rdfs:comment | 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 nm × 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: In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator (en) |
| rdfs:label | Commutation matrix (en) |
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