分块矩阵

加法

设 A=(A11⋯A1t⋮⋱⋮As1⋯Ast)A=\begin{pmatrix}A_{11}&\cdots&A_{1t}\\\vdots&\ddots&\vdots\\A_{s1}&\cdots&A_{st}\end{pmatrix}，
B=(B11⋯B1t⋮⋱⋮Bs1⋯Bst)B = \begin{pmatrix}B_{11}&\cdots&B_{1t}\\\vdots&\ddots&\vdots\\B_{s1}&\cdots&B_{st}\end{pmatrix}，则

A+B=(A11+B11⋯A1t+B1t⋮⋱⋮As1+Bs1⋯Ast+Bst)A + B = \begin{pmatrix}A_{11} + B_{11}&\cdots&A_{1t}+B_{1t}\\\vdots&\ddots&\vdots\\A_{s1}+B_{s1}&\cdots&A_{st}+B_{st}\end{pmatrix}

数乘

设 A=(A11⋯A1t⋮⋱⋮As1⋯Ast)A=\begin{pmatrix}A_{11}&\cdots&A_{1t}\\ \vdots&\ddots&\vdots\\A_{s1}&\cdots&A_{st}\end{pmatrix}，则

kA=(kA11⋯kA1t⋮⋱⋮kAs1⋯kAst)kA = \begin{pmatrix}kA_{11}&\cdots&kA_{1t}\\\vdots&\ddots&\vdots\\kA_{s1}&\cdots&kA_{st}\end{pmatrix}

乘法

设 A=(A11⋯A1t⋮⋱⋮As1⋯Ast)A=\begin{pmatrix}A_{11}&\cdots&A_{1t}\\ \vdots&\ddots&\vdots\\A_{s1}&\cdots&A_{st}\end{pmatrix}，B=(B11⋯B1r⋮⋱⋮Bt1⋯Btr)B = \begin{pmatrix}B_{11}&\cdots&B_{1r}\\\vdots&\ddots&\vdots\\B_{t1}&\cdots&B_{tr}\end{pmatrix}，则

AB=(C11⋯C1r⋮⋱⋮Cs1⋯Csr)AB = \begin{pmatrix}C_{11}&\cdots&C_{1r}\\\vdots&\ddots&\vdots\\C_{s1}&\cdots&C_{sr}\end{pmatrix}

其中 Cij=∑k=1tAikBkjC_{ij}=\sum_{k=1}^tA_{ik}B_{kj}

幂

设 A=(A1A2⋱As)A = \begin{pmatrix} A_1 & & & \\ & A_2 & & \\ & & \ddots &\\ & & & A_s \end{pmatrix}，则

Am=(A1mA2m⋱Asm) A^m = \begin{pmatrix} A_1^m & & &\\ & A_2^m & &\\ & & \ddots &\\ & & & A_s^m\\ \end{pmatrix}

转置

设 A=(A11A12⋯A1tA21A22⋯A2t⋮⋮⋱⋮As1As2⋯Ast)A=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1t}\\A_{21}&A_{22}&\cdots&A_{2t}\\ \vdots&\vdots&\ddots&\vdots\\A_{s1}&A_{s2}&\cdots&A_{st}\end{pmatrix}，则

A′=(A11′A21′⋯As1′A12′A22′⋯As2′⋮⋮⋱⋮A1t′A2t′⋯Ast′)A' = \begin{pmatrix}A_{11}^\prime&A_{21}^\prime&\cdots&A_{s1}^\prime\\A_{12}^\prime&A_{22}^\prime&\cdots&A_{s2}^\prime\\ \vdots&\vdots&\ddots&\vdots\\A_{1t}^\prime&A_{2t}^\prime&\cdots&A_{st}^\prime\end{pmatrix}

行列式

∣A11A12⋯A1s0A22⋯A2s⋮⋮⋱⋮00⋯Ass∣=∣A11∣∣A22∣⋯∣Ass∣\begin{vmatrix} A_{11} &A_{12} &\cdots &A_{1s}\\ 0 &A_{22} &\cdots &A_{2s}\\ \vdots &\vdots &\ddots &\vdots\\ 0 &0 &\cdots &A_{ss} \end{vmatrix} = \left\vert A_{11}\right\vert \left\vert A_{22}\right\vert\cdots\left\vert A_{ss}\right\vert

∣A110⋯0A21A22⋯0⋮⋮⋱⋮As1As2⋯Ass∣=∣A11∣∣A22∣⋯∣Ass∣\begin{vmatrix} A_{11} & 0 & \cdots & 0\\ A_{21} & A_{22} & \cdots & 0\\ \vdots &\vdots &\ddots &\vdots\\ A_{s1} & A_{s2} & \cdots & A_{ss} \end{vmatrix} = \left\vert A_{11}\right\vert \left\vert A_{22}\right\vert\cdots\left\vert A_{ss}\right\vert

∣A1A2⋱As∣=∣A1∣∣A2∣⋯∣As∣\begin{vmatrix} A_{1} & & & \\ & A_{2} & & \\ & & \ddots & \\ & & & A_{s} \end{vmatrix} = \left\vert A_{1}\right\vert \left\vert A_{2}\right\vert\cdots\left\vert A_{s}\right\vert

逆

(A1A2⋱As)−1=(A1−1A2−1⋱As−1) \begin{pmatrix} A_1 & & &\\ & A_2 & &\\ & & \ddots &\\ & & &A_s \end{pmatrix}^{-1} = \begin{pmatrix} A_1^{-1} & & &\\ & A_2^{-1} & &\\ & & \ddots &\\ & & &A_s^{-1} \end{pmatrix}

(A1A2⋱As)−1=(As−1As−1−1⋱A1−1) \begin{pmatrix} & & &A_1 \\ & & A_2 &\\ &\ddots & &\\ A_s& & & \end{pmatrix}^{-1} = \begin{pmatrix} & & &A_s^{-1}\\ && A_{s-1}^{-1} &\\ & \ddots && \\ A_1^{-1}& & & \end{pmatrix}

降阶公式

设 AA 为 m×nm\times n 矩阵，BB 是 n×mn\times m 矩阵，m>nm>n，λ\lambda 是任意数，则

∣λEm−AB∣=λm−n∣λEn−BA∣\left\vert\lambda E_m - AB\right\vert = \lambda^{m-n}\left\vert\lambda E_n - BA\right\vert

矩阵的秩

定义

矩阵 AA 的非零子式的最高阶数叫作矩阵 AA 的秩。

性质

设 AA 为 m×nm\times n 矩阵，BB 为 n×pn\times p 矩阵，则

  • 0≤R(A)≤min⁡{m,n}0 \le R(A) \le \min\{m, n\}
  • R(A′)=R(A)R(A') = R(A)
  • R(kA)={0k=0R(A)k≠0R(kA) = \begin{cases}0&k=0\\R(A)&k\ne0\end{cases}
  • R(A1)≤R(A)R(A_1) \le R(A)，其中 A1A_1 为 AA 的任意一个子矩阵。
  • R(A00B)=R(A)+R(B)R\begin{pmatrix}A&0\\0&B\end{pmatrix} = R(A) + R(B)
  • R(AC0B)≥R(A)+R(B)R\begin{pmatrix}A&C\\0&B\end{pmatrix} \ge R(A) + R(B)
  • R(A∣B)≤R(A)+R(B)R(A \mid B) \le R(A) + R(B)
  • R(A+B)≤R(A)+R(B)R(A + B) \le R(A) + R(B)
  • R(AB)≤min⁡{R(A),R(B)}R(AB) \le \min\{R(A), R(B)\}
  • R(AB)≥R(A)+R(B)−nR(AB) \ge R(A) + R(B) - n