Block Diagonal Matrix Transpose

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$A = \begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1q} \\ A_{21}&A_{22}&\cdots&A_{2q} \\ \vdots&\vdots&\ddots&\vdots \\ A_{q1}&A_{q2}&\cdots&A_{qq} \end{bmatrix}$

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$B = \begin{bmatrix} B_{11}&B_{12}&\cdots&B_{1q} \\ B_{21}&B_{22}&\cdots&B_{2q} \\ \vdots&\vdots&\ddots&\vdots \\ B_{q1}&B_{q2}&\cdots&B_{qq} \end{bmatrix} \qquad I = \begin{bmatrix} I_{11}&I_{12}&\cdots&I_{1q} \\ I_{21}&I_{22}&\cdots&I_{2q} \\ \vdots&\vdots&\ddots&\vdots \\ I_{q1}&I_{q2}&\cdots&I_{qq} \end{bmatrix}$

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$B = \begin{bmatrix} B_{1}&0&\cdots&0 \\ 0&B_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&B_{q} \end{bmatrix}$

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$A = \begin{bmatrix} 0&x&0&0&0 \\ x&0&0&0&0 \\ 0&x&0&0&0 \\ 0&0&0&0&0 \\ 0&0&0&0&x \end{bmatrix}$

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$A'_{\alpha} = \begin{bmatrix} a_{w_\alpha+1, w_\alpha+1}&a_{w_\alpha+1,w_\alpha + 2}&\cdots&a_{w_\alpha+1,w_{\alpha+1}} \\ a_{w_\alpha+2, w_\alpha+1}&a_{w_\alpha+2,v_\alpha+2}&\cdots&a_{w_\alpha+2,w_{\alpha+1}} \\ \vdots&\vdots&\ddots&\vdots \\ a_{w_{\alpha+1}, w_\alpha+1}&a_{w_{\alpha+1},w_\alpha+2}&\cdots&a_{w_{\alpha+1},w_{\alpha+1}} \end{bmatrix}$

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$C = \begin{bmatrix} C_{1}&0&\cdots&0 \\ 0&C_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&C_{q} \end{bmatrix} = \begin{bmatrix} A_{1}B_{1}&0&\cdots&0 \\ 0&A_{2}B_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&A_{q}B_{q} \end{bmatrix}$

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$C = \begin{bmatrix} C_{1}&0&\cdots&0 \\ 0&C_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&C_{q} \end{bmatrix}$

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$A = \begin{bmatrix} a_1&0&\cdots&0 \\ 0&a_2&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&a_n \end{bmatrix} \qquad B = \begin{bmatrix} b_1&0&\cdots&0 \\ 0&b_2&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&b_n \end{bmatrix}$

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$C = AB = \begin{bmatrix} C_{1}&0&\cdots&0 \\ 0&C_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&C_{q} \end{bmatrix} = \begin{bmatrix} A_{1}B_{1}&0&\cdots&0 \\ 0&A_{2}B_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&A_{q}B_{q} \end{bmatrix}$

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Block Diagonal Matrix Transpose

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