文章目录
- 1. 矩阵A分解
1. 矩阵A分解
对于矩阵A来说,我们有常见矩阵分解:
A = L U , A = Q R , A = X Λ X − 1 , A = Q Λ Q T ; A = Q S , A = S V D \begin{equation} A=LU,A=QR,A=X\Lambda X^{-1},A=Q\Lambda Q^T;A=QS,A=SVD \end{equation} A=LU,A=QR,A=XΛX−1,A=QΛQT;A=QS,A=SVD
1.1 A = L U A=LU A=LU
我们矩阵A可以进行LU分解,那么L表示下三角矩阵,U表示上三角矩阵,对于L上三角矩阵来说;举例如下:
A = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] = [ l 11 0 0 l 21 l 22 0 l 31 l 32 l 33 ] [ u 11 u 12 u 13 0 u 22 u 23 0 0 u 33 ] \begin{equation} A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\\\ a_{21}&a_{22}&a_{23}\\\\ a_{31}&a_{32}&a_{33} \end{bmatrix}=\begin{bmatrix} l_{11}&0&0\\\\ l_{21}&l_{22}&0\\\\ l_{31}&l_{32}&l_{33} \end{bmatrix}\begin{bmatrix} u_{11}&u_{12}&u_{13}\\\\ 0&u_{22}&u_{23}\\\\ 0&0&u_{33} \end{bmatrix} \end{equation} A= a11a21a31a12a22a32a13a23a33 = l11l21l310l22l3200l33 u1100u12u220u13u23u33