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Let the operator vec($A$) unroll all the elements of $A$ into a single column vector in column-major order. Then, the elements of vec($A^T$) are a permutation of the elements of vec($A$). If I want to write this permutation as a matrix-vector product, I get

vec($A^T)$ = $P$ vec($A$).

I'm looking for a common name and/or symbol for the "vec transposition permutation matrix" $P$?

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    $\begingroup$ Commutation matrix $\endgroup$
    – Rodrigo de Azevedo
    Commented Aug 11, 2017 at 4:30
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    $\begingroup$ In addition to the name "commutation matrix", this sometimes goes by the name "swap gate", and it appears lots in quantum computing. $\endgroup$
    – Nathaniel Johnston
    Commented Aug 12, 2017 at 3:15

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This matrix is known as the commutation matrix. For more details, please see Chapter 3, Section 7 of: Matrix differential calculus by Magnus and Neudecker.

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