Bordered Block Diagonal Form

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Figure 3 from Stabilized Bordered Block Diagonal Form for Solving ...

Ordering to singly bordered block diagonal form

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

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$A = \begin{bmatrix} A_{1}&0&\cdots&0 \\ 0&A_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&A_{r} \end{bmatrix}$

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$A^{-1} = \begin{bmatrix} B_{11}&0&\cdots&0 \\ 0&B_{22}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&B_{qq} \end{bmatrix} = \begin{bmatrix} A^{-1}_{1}&0&\cdots&0 \\ 0&A^{-1}_{2}&\cdots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\cdots&A^{-1}_{q} \end{bmatrix}$

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Doubly bordered block-diagonal matrix | Download Scientific Diagram

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Bordered Block Diagonal Form

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