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| In [[mathematics]], especially in [[linear algebra]] and [[Matrix (mathematics)|matrix theory]], the '''commutation matrix''' is used for transforming the vectorized form of a [[matrix (mathematics)|matrix]] into the vectorized form of its [[transpose]]. Specifically, the commutation matrix '''K'''<sup>(m,n)</sup> is the ''mn × mn'' matrix which, for any ''m × n'' matrix '''A''', transforms vec('''A''') into vec('''A'''<sup>T</sup>):
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| :'''K'''<sup>(m,n)</sup> vec('''A''') = vec('''A'''<sup>T</sup>) .
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| Here vec('''A''') is the ''mn × 1'' [[column vector]] obtain by stacking the columns of '''A''' on top of one another:
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| :vec('''A''') = [ '''A'''<sub>1,1</sub>, ..., '''A'''<sub>m,1</sub>, '''A'''<sub>1,2</sub>, ..., '''A'''<sub>m,2</sub>, ..., '''A'''<sub>1,n</sub>, ..., '''A'''<sub>m,n</sub> ]<sup>T</sup>
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| where '''A''' = ['''A'''<sub>i,j</sub>].
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| The commutation matrix is a special type of [[permutation matrix]], and is therefore [[orthogonal matrix|orthogonal]]. It is also an [[involution (mathematics)#Linear Algebra|involution]] and [[symmetric matrix|symmetric]].
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| 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''',
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| :'''K'''<sup>(r,m)</sup>('''A''' <math>\otimes</math> '''B''')'''K'''<sup>(n,q)</sup> = '''B''' <math>\otimes</math> '''A'''. | |
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| An explicit form for the commutation matrix is as follows: if '''e'''<sub>r,j</sub> denotes the j-th canonical vector of dimension ''r'' (i.e. the vector with 1 in the j-th coordinate and 0 elsewhere) then
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| :'''K'''<sup>(r,m)</sup> = <math>\sum_{i=1}^{r}</math><math>\sum_{j=1}^{m}</math>'''e'''<sub>r,i</sub>'''e'''<sub>m,j</sub><sup>T</sup><math>\otimes</math>'''e'''<sub>m,j</sub>'''e'''<sub>r,i</sub><sup>T</sup>.
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| ==References== | |
| Jan R. Magnus and Heinz Neudecker (1988), ''Matrix Differential Calculus with Applications in Statistics and Econometrics'', Wiley.
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| [[Category:Linear algebra]]
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| [[Category:Matrices]]
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| {{Linear-algebra-stub}}
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