Multiplying Block Matrices

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$= \begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1p} \\ a_{21}&a_{22}&\cdots&a_{2p} \\ \vdots&\vdots&\ddots&\vdots \\ a_{m1}&a_{m2}&\cdots&a_{mp} \end{bmatrix} = A$

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$B_{\alpha \beta} = \begin{bmatrix} b_{w_\alpha+1 ,v_\beta+1}&b_{w_\alpha+1,v_\beta+2}&\cdots&b_{w_\alpha+1,v_{\beta+1}} \\ b_{w_\alpha+2, v_\beta+1}&b_{w_\alpha+2,v_\beta+2}&\cdots&b_{w_\alpha+2,v_{\beta+1}} \\ \vdots&\vdots&\ddots&\vdots \\ b_{w_{\alpha+1}, v_\beta+1}&b_{w_{\alpha+1},v_\beta+2}&\cdots&b_{w_{\alpha+1},v_{\beta+1}} \end{bmatrix}$

Multiplying block matrices | My Math Homework

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

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$= \begin{bmatrix} a_{(\alpha-1)+1,(\beta-1)+1}&\cdots&a_{(\alpha-1)+1,(\beta+1)-1} \\ \vdots&\vdots&\ddots&\vdots \\ a_{(\alpha+1)-1,(\beta-1)+1}&\cdots&a_{(\alpha+1)-1,(\beta+1)-1} \end{bmatrix} = \begin{bmatrix} a_{\alpha \beta} \end{bmatrix}$

Multiplying block matrices | My Math Homework

$C = \begin{bmatrix} C_{11}&C_{12}&\cdots&C_{1r} \\ C_{21}&C_{22}&\cdots&C_{2r} \\ \vdots&\vdots&\ddots&\vdots \\ C_{q1}&C_{q2}&\cdots&C_{qr} \end{bmatrix}$

Multiplying block matrices | My Math Homework

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

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$v = \begin{bmatrix} v_1&v_2&\cdots&v_{r+1} \end{bmatrix}$

Multiplying block matrices | My Math Homework

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