$\require{cancel} \newcommand{\Ket}[1]{\left|{#1}\right\rangle} \newcommand{\Bra}[1]{\left\langle{#1}\right|} \newcommand{\Braket}[1]{\left\langle{#1}\right\rangle} \newcommand{\Rsr}[1]{\frac{1}{\sqrt{#1}}} \newcommand{\RSR}[1]{1/\sqrt{#1}} \newcommand{\Verti}{\rvert} \newcommand{\HAT}[1]{\hat{\,#1~}} \DeclareMathOperator{\Tr}{Tr}$

Block Matrix

^{First created in July 2018}

Ref: Block matrix

$M =\begin{bmatrix}A_1&B_1\\C_1&D_1\end{bmatrix} \begin{bmatrix}A_2&B_2\\C_2&D_2\end{bmatrix} =\begin{bmatrix}A_1A_2+B_1C_2&A_1B_2+B_1D_2\\C_1A_2+D_1C_2&C_1B_2+D_1D_2\end{bmatrix} .$

$A_1A_2+B_1C_2 =\begin{bmatrix} a_{1_{11}}a_{2_{11}}+a_{1_{12}}a_{2_{21}}& a_{1_{11}}a_{2_{12}}+a_{1_{12}}a_{2_{22}}\\ a_{1_{21}}a_{2_{11}}+a_{1_{22}}a_{2_{21}}& a_{1_{21}}a_{2_{12}}+a_{1_{22}}a_{2_{22}} \end{bmatrix} +\begin{bmatrix} b_{1_{11}}c_{2_{11}}+b_{1_{12}}c_{2_{21}}& b_{1_{11}}c_{2_{12}}+b_{1_{12}}c_{2_{22}}\\ b_{1_{21}}c_{2_{11}}+b_{1_{22}}c_{2_{21}}& b_{1_{21}}c_{2_{12}}+b_{1_{22}}c_{2_{22}} \end{bmatrix} .$

$A_1B_2+B_1D_2 =\begin{bmatrix} a_{1_{11}}b_{2_{11}}+a_{1_{12}}b_{2_{21}}& a_{1_{11}}b_{2_{12}}+a_{1_{12}}b_{2_{22}}\\ a_{1_{21}}b_{2_{11}}+a_{1_{22}}b_{2_{21}}& a_{1_{21}}b_{2_{12}}+a_{1_{22}}b_{2_{22}} \end{bmatrix} +\begin{bmatrix} b_{1_{11}}d_{2_{11}}+b_{1_{12}}d_{2_{21}}& b_{1_{11}}d_{2_{12}}+b_{1_{12}}d_{2_{22}}\\ b_{1_{21}}d_{2_{11}}+b_{1_{22}}d_{2_{21}}& b_{1_{21}}d_{2_{12}}+b_{1_{22}}d_{2_{22}} \end{bmatrix} .$

$M =\begin{bmatrix} a_{1_{11}}a_{2_{11}}+a_{1_{12}}a_{2_{21}}+ b_{1_{11}}c_{2_{11}}+b_{1_{12}}c_{2_{21}}& a_{1_{11}}a_{2_{12}}+a_{1_{12}}a_{2_{22}}+ b_{1_{11}}c_{2_{12}}+b_{1_{12}}c_{2_{22}}& a_{1_{11}}b_{2_{11}}+a_{1_{12}}b_{2_{21}}+ b_{1_{11}}d_{2_{11}}+b_{1_{12}}d_{2_{21}}& a_{1_{11}}b_{2_{12}}+a_{1_{12}}b_{2_{22}}+ b_{1_{11}}d_{2_{12}}+b_{1_{12}}d_{2_{22}}\\ a_{1_{21}}a_{2_{11}}+a_{1_{22}}a_{2_{21}}+ b_{1_{21}}c_{2_{11}}+b_{1_{22}}c_{2_{21}}& a_{1_{21}}a_{2_{12}}+a_{1_{22}}a_{2_{22}}+ b_{1_{21}}c_{2_{12}}+b_{1_{22}}c_{2_{22}}& a_{1_{21}}b_{2_{11}}+a_{1_{22}}b_{2_{21}}+ b_{1_{21}}d_{2_{11}}+b_{1_{22}}d_{2_{21}}& a_{1_{21}}b_{2_{12}}+a_{1_{22}}b_{2_{22}}+ b_{1_{21}}d_{2_{12}}+b_{1_{22}}d_{2_{22}}\\ C_1A_2+D_1C_2&&C_1B_2+D_1D_2& \end{bmatrix} .$

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The RHS is $\begin{bmatrix} a_{1_{11}}a_{2_{11}}+b_{1_{11}}c_{2_{11}}& a_{1_{12}}a_{2_{21}}+b_{1_{12}}c_{2_{21}}& a_{1_{11}}b_{2_{11}}+b_{1_{11}}d_{2_{11}}& a_{1_{12}}b_{2_{21}}+b_{1_{12}}d_{2_{21}}\\ \ldots\\ \ldots \end{bmatrix} .$