6.3. 广义逆计算

6.3.1. Hermite标准形计算 {1}-逆 和 {1,2}-逆

6.3.2. 满秩分解法

设 \({\bm A} \in {\mathbb C}_r^{m\times n} (r>0)\) , 若满秩分解为 \({\bm A}_r^{m\times n} = {\bm F}_r^{m\times r} {\bm G}_r^{r\times n}\) , 则有

1. \({\bm G}^{(1)}{\bm F}^{(i)} \in {\bm A}\{i\} (i=1,2,3)\)

2. \({\bm G}^{(i)}{\bm F}^{(1)} \in {\bm A}\{i\} (i=1,2,4)\)

3. \({\bm G}^{(1)}{\bm F}^{+} \in {\bm A}\{1,2,3\}\)

4. \({\bm G}^{+}{\bm F}^{(1)} \in {\bm A}\{1,2,4\}\)

5. \({\bm A}^+ = {\bm G}^+{\bm F}^{(1,3)} = {\bm G}^{(1,4)}{\bm F}^+\)

6. \({\bm A}^+ = {\bm G}^+{\bm F}^+ = {\bm G}^H({\bm G}{\bm G}^H)^{-1} ({\bm F}^H {\bm F})^{-1}{\bm F}^H = {\bm G}^H({\bm F}^H{\bm F}{\bm G}{\bm G}^H)^{-1}{\bm F}^H = {\bm G}^H({\bm F}^H{\bm A}{\bm G}^H)^{-1}{\bm F}^H\)