或る大学院生のメモ帳: Inverse of A (matrix): Analytical Solution

2012年5月17日木曜日

Inverse of A (matrix): Analytical Solution

$\mathbf{A}^{-1}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}^{\mathrm{T}}\right)_{ij}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}_{ji}\right)={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}} \begin{pmatrix} \mathbf{C}_{11} & \mathbf{C}_{21} & \cdots & \mathbf{C}_{n1} \\ \mathbf{C}_{12} & \mathbf{C}_{22} & \cdots & \mathbf{C}_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{C}_{1n} & \mathbf{C}_{2n} & \cdots & \mathbf{C}_{nn} \\ \end{pmatrix}$

where C(i,j) is the cofactor of A(i,j) such that

$C_{ij}=(-1)^{i+j} M_{ij} \,$

M(i,j) is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column. For example,

$M_{23} = \begin{vmatrix} b_{11} & b_{12} & \Box \\ \Box & \Box & \Box \\ b_{31} & b_{32} & \Box \\ \end{vmatrix}$

yields

$M_{23} = \begin{vmatrix} b_{11} & b_{12} \\ b_{31} & b_{32} \\ \end{vmatrix} = b_{11}b_{32} - b_{31}b_{12}$

http://en.wikipedia.org/wiki/Cramer%27s_rule
http://en.wikipedia.org/wiki/Invertible_matrix#Methods_of_matrix_inversion
http://en.wikipedia.org/wiki/Matrix_of_cofactors

或る大学院生のメモ帳

2012年5月17日木曜日

Inverse of A (matrix): Analytical Solution

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