Block Matrix Diagonalization

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Matrix Block Diagonalization Theorem – LUDQJ

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Determinant of a block matrix

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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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$A = \begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1r} \\ A_{21}&A_{22}&\cdots&A_{2r} \\ \vdots&\vdots&\ddots&\vdots \\ A_{r1}&A_{r2}&\cdots&A_{rr} \end{bmatrix} \qquad A_{\alpha \beta} = 0 \text{ if } \alpha \ne \beta$

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Matrix diagonalization

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

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Determinant of a block matrix

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Diagonal Matrix Diagonal Matrices

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Diagonal Matrix Diagonal Matrices

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Diagonal Matrix Diagonal Matrices

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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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Diagonal Matrix

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Diagonal Matrix Diagonal Matrices

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this figure shows how a block-dense matrix A can be factorized into a ...

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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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Diagonal Matrix Diagonal Matrices

$C = AB = \begin{bmatrix} C_{1}&0&\cdots&0 \\ 0&C_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&C_{n} \end{bmatrix} = \begin{bmatrix} a_1b_1&0&\cdots&0 \\ 0&a_2b_2&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&a_nb_n \end{bmatrix}$