Block Matrix Inverse

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Matrix Inverse in Block Form

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

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

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

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Matrix Inverse in Block Form

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

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Block Matrix Inverse | PDF

Matrix Inverse in Block Form

Transformation from blocked impedance matrix to block inverse ...

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

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Matrix Inverse in Block Form

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Block Matrix Inverse

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