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Let $\mathbf{A}=\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ A_{m1} & A_{m2} & \cdots & A_{mm} \end{bmatrix}$ be a block matrix where each $A_{ij}$, $1\leq i,j\leq m$, is an

$n\times n$ matrix. Define operator $VB$ by

$VB(\mathbf{A})=\begin{bmatrix} A_{11} \\ A_{21} \\ \vdots \\ A_{m1} \\ A_{12} \\ \vdots \\ A_{mm} \end{bmatrix}$

What is called operator $VB$? Is there any reference about its properties?

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    $\begingroup$ This is not exactly an operator: this is an isomorphism between two distinct vector spaces. What kind of properties do you want to know? $\endgroup$ – Alex Degtyarev Jul 12 '15 at 16:40
  • $\begingroup$ We have vec operator of a scalar matrix. I want some proposition related with famous matrix operator (for example, usual and kronecker product with other matrices, etc) $\endgroup$ – savalan savalan Jul 12 '15 at 17:11
  • $\begingroup$ My friend and collaborator A. I. Solomon used to call this operation "the stringing" (pronounce like "a stringed instrument"), he said, you stringe them. I met also this operation in genetic algorithms (I'll try to find a reference) $\endgroup$ – Duchamp Gérard H. E. Jul 12 '15 at 18:55
  • $\begingroup$ I would say $blkvec$ following the $diag$ case $\endgroup$ – percusse Jul 31 '15 at 0:17
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the inverse of $VB$ maps a tensor onto a matrix; this operation is called matricization (or unfolding or flattening), see this paper by Lars Eldén and Berkant Savas (2009).

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