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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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It's called the vec operator.

Its most interesting property is with respect to the Kronecker product $$ {\rm vec}(AXB) = (B^T\otimes A)\,{\rm vec}(X) $$

Update

I missed the fact that $A$ was specified to be a block matrix. The "block-vec" operation defined in this paper by Ignacio Ojeda is what you are looking for.

In Ojeda's notation, the operation you have defined is $\,\,{\rm vec}^{(1\times n)}(A)$

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