1 秩与维数

1. ${\displaystyle \mathrm{rank}A=\dim \mu (A)}$.

2. ${\displaystyle n=\dim \mathrm{Im}(A)+\dim \mathrm{Ker}(A)}$.

3. ${\displaystyle \mathrm{rank}(AB)=\mathrm{rank}B-\dim (\mathrm{Im}(B)\cap \mathrm{Ker}(A))}$.

4. ${\displaystyle A}$ 满列秩等价于 ${\displaystyle \mathrm{\exists }{A}_{\mathrm{l}}}$: ${\displaystyle A_{\mathrm{l}}A=I_n}$; ${\displaystyle A}$ 满行秩等价于 ${\displaystyle \mathrm{\exists }{A}_{\mathrm{r}}}$: ${\displaystyle A{A}_{\mathrm{r}}=I_m}$.

2 子空间

1. ${\displaystyle \dim ({\mathcal{L}}_1+{\mathcal{L}}_2)=\dim {\mathcal{L}}_1+\dim {\mathcal{L}}_2-\dim ({\mathcal{L}}_1\cap {\mathcal{L}}_2)}$.
   • ${\displaystyle \dim ({\mathcal{L}}_1\oplus {\mathcal{L}}_2)=\dim {\mathcal{L}}_1+\dim {\mathcal{L}}_2}$.
2. ${\displaystyle {\mathcal{L}}_1\perp {\mathcal{L}}_{\mathcal{2}}\Rightarrow {\mathcal{L}}_1+{\mathcal{L}}_2={\mathcal{L}}_1\oplus {\mathcal{L}}_2}$.
3. ${\displaystyle \mathcal{L}\perp {\mathcal{L}}^{\perp }}$.
4. ${\displaystyle {\mathcal{L}}_1\subset {\mathcal{L}}_2⟺{\mathcal{L}}_2^{\perp }\subset {\mathcal{L}}_1^{\perp }}$.

3 线性方程的相容

线性方程 ${\displaystyle Ax=b}$ 相容/有解的充要条件是以下之一:

• ${\displaystyle b\in \mathrm{Im}(A)}$.
• ${\displaystyle c^{\mathrm{T}}A=0\Rightarrow c^{\mathrm{T}}b=0}$.
• ${\displaystyle \mathrm{rank}((Ab))=\mathrm{rank}A}$.

4 特征值

设 ${\displaystyle A}$ 的特征值为 ${\displaystyle {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n}$.

1. 设 ${\displaystyle A^{\mathrm{T}}=A}$, 则 ${\displaystyle \mathrm{\exists }C}$ 正交: ${\displaystyle C^{\mathrm{T}}AC=\mathrm{\Lambda }=\mathrm{diag}({\lambda }_1,\cdots ,{\lambda }_n)}$. 这里 ${\displaystyle c_j}$ 是 ${\displaystyle A}$ 对应于 ${\displaystyle {\lambda }_j}$ 的单位特征向量; ${\displaystyle A=\sum _{j=1}^n{\lambda }_j{c}_j{c}_j^{\mathrm{T}}}$.

2. 设 ${\displaystyle A^{\mathrm{T}}=A}$. 则 $${\lambda }_1=\underset{||x||=1}{max}{x}^{\mathrm{T}}Ax,{\lambda }_i=\underset{||x||=1,x\perp \mu (c_1,\cdots ,c_{i-1})}{max}{x}^{\mathrm{T}}Ax.$$

3. 设 ${\displaystyle A^{\mathrm{T}}=A}$, ${\displaystyle U_{n\times p}}$ 列正交 (${\displaystyle U^{\mathrm{T}}U=I_p}$), 则 $$\underset{U^{\mathrm{T}}U=I_p}{max}\mathrm{tr}(U^{\mathrm{T}}AU)=\sum _{i=1}^p{\lambda }_i.$$

4. 设 ${\displaystyle A^{\mathrm{T}}=A}$, 下列命题等价:

   • ${\displaystyle A}$ 的特征值 ${\displaystyle {\lambda }_1\ge \cdots \ge {\lambda }_n\ge 0}$.
   • ${\displaystyle x^{\mathrm{T}}Ax\ge 0}$, ${\displaystyle \mathrm{\forall }x\in {\mathbb{R}}^n}$, ${\displaystyle x\ne 0}$.
   • ${\displaystyle \mathrm{\exists }B}$: ${\displaystyle A=B{B}^{\mathrm{T}}}$. (即 ${\displaystyle A}$ 非负定, 记为 ${\displaystyle A\ge 0}$)

5. ${\displaystyle A\ge 0\Rightarrow }$ ${\displaystyle \mathrm{\exists }!A^{\frac{1}{2}}=C{\mathrm{\Lambda }}^{\frac{1}{2}}{C}^{\mathrm{T}}}$: ${\displaystyle A=A^{\frac{1}{2}}{A}^{\frac{1}{2}}}$. ${\displaystyle C}$ 见 4.1.

6. ${\displaystyle B_{n\times m}{B}^{\mathrm{T}}=D_{n\times p}{D}^{\mathrm{T}}}$ ${\displaystyle (p\le m)}$ ${\displaystyle ⟺}$ ${\displaystyle B=(C0)U}$, 其中 ${\displaystyle U_{m\times m}}$ 正交.

7. ${\displaystyle \mathrm{rank}(B{B}^{\mathrm{T}})=\mathrm{rank}B}$.

5 分快矩阵

剖分 $$A_{m\times n}=\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1q}\\ ⋮& & ⋮\\ A_{p1}& \cdots & A_{pq}\end{array}\right),B_{n\times k}=\left(\begin{array}{ccc}{B}_{11}& \cdots & B_{1r}\\ ⋮& & ⋮\\ B_{q1}& \cdots & B_{qr}\end{array}\right),$$ 且 ${\displaystyle AB=C=\left(\begin{array}{ccc}{C}_{11}& \cdots & C_{1r}\\ ⋮& & ⋮\\ C_{p1}& \cdots & C_{pr}\end{array}\right)}$, 其中 ${\displaystyle C_{ij}=\sum _{l=1}^q{A}_{il}{B}_{lj}}$.

1. 设 ${\displaystyle A=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)}$, 其中 ${\displaystyle A_{ii}}$ 是 ${\displaystyle p_i}$ 阶方阵.

   • 如果 ${\displaystyle A_{11}}$ 可逆, ${\displaystyle \det A=det{A}_{11}\cdot det(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12})}$.
   • 如果 ${\displaystyle A_{22}}$ 可逆, ${\displaystyle detA=det{A}_{22}\cdot det(A_{11}-A_{12}{A}_{22}^{-1}{A}_{21})}$.

   ${\displaystyle A_{11}}$ 可逆时 $$\left(\begin{array}{cc}{I}_{p1}& 0\\ -A_{21}{A}_{11}^{-1}& I_{p2}\end{array}\right)\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\left(\begin{array}{cc}{I}_{p1}& -A_{11}^{-1}{A}_{12}\\ 0& I_{p2}\end{array}\right)=\left(\begin{array}{cc}{A}_{11}& 0\\ 0& A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}\end{array}\right).$$ 故 ${\displaystyle A}$ 可逆时 $$A^{-1}=\left[\left(\begin{array}{cc}{I}_{p1}& 0\\ A_{21}{A}_{11}^{-1}& I_{p2}\end{array}\right)\left(\begin{array}{cc}{A}_{11}& 0\\ 0& A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}\end{array}\right)\left(\begin{array}{cc}{I}_{p1}& A_{11}^{-1}{A}_{12}\\ 0& I_{p2}\end{array}\right)\right].$$

2. 记 ${\displaystyle A^{-1}=\left(\begin{array}{cc}{A}^{11}& A^{12}\\ A^{21}& A^{22}\end{array}\right)}$, 有$$\begin{array}{rl}{A}^{11}& =A_{11}^{-1}+A_{11}^{-1}{A}_{12}(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}{)}^{-1}{A}_{21}{A}_{11}^{-1},\\ A^{12}& =-A_{11}^{-1}{A}_{12}(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}{)}^{-1},\\ A^{21}& =-(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}{)}^{-1}{A}_{21}{A}_{11}^{-1},\\ A^{22}& =(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}{)}^{-1}.\end{array}$$

3. 设 ${\displaystyle A>0}$. 则 ${\displaystyle A^{11}-A_{11}^{-1}\ge 0}$, 且 $$A^{-1}-\left(\begin{array}{cc}{A}_{11}^{-1}& 0\\ 0& 0\end{array}\right)=\left(\begin{array}{c}-A_{11}^{-1}{A}_{12}\\ I\end{array}\right)A^{22}{\left(\begin{array}{c}-A_{11}^{-1}{A}_{12}\\ I\end{array}\right)}^{\mathrm{T}}\ge 0.$$

4. ${\displaystyle A_{m\times n}}$, ${\displaystyle B_{n\times m}}$, 则 ${\displaystyle det(\lambda I_m-AB)={\lambda }^{-n}det(\lambda I_n-BA)}$, 从而 ${\displaystyle AB,BA}$ 有相同的非零特征值.

6 投影

1. ${\displaystyle A}$ 满列秩, ${\displaystyle \mathrm{Im}(A)=\mathcal{L}}$, 则 ${\displaystyle P_{\mathcal{L}}=A(A^{\mathrm{T}}A{)}^{-1}{A}^{\mathrm{T}}}$. 这
   这说明 ${\displaystyle P_{\mathcal{L}}}$ 此时唯一确定. 可以记 ${\displaystyle P_{\mathcal{L}}=P_A}$.

2. ${\displaystyle P}$ 是到 ${\displaystyle \mathrm{Im}(P)}$ 的正投影阵的充要条件是以下任一:

   • ${\displaystyle P}$ 是对称幂等阵.
   • ${\displaystyle ||x-Px|{|}^2=\underset{t\in {\mathbb{R}}^n}{min}||x-Pt|{|}^2}$, ${\displaystyle \mathrm{\forall }x\in {\mathbb{R}}^n}$.
   • ${\displaystyle P^2=P}$, 且 ${\displaystyle ||Px||\le ||x||}$, ${\displaystyle \mathrm{\forall }x\in {\mathbb{R}}^n}$.
   • ${\displaystyle \mathrm{\exists }U}$ 正交: ${\displaystyle P=U\left(\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right)U^{\mathrm{T}}}$, ${\displaystyle r=\mathrm{rank}P}$. 记 ${\displaystyle U=(U_1^r{U}_2)}$, 得 ${\displaystyle P=U_1{U}_1^{\mathrm{T}}}$, 且 ${\displaystyle U_1^{\mathrm{T}}{U}_1=I_r}$.
   • ${\displaystyle I-P}$ 对称幂等.

3. ${\displaystyle P_A}$ 是到 ${\displaystyle \mathrm{Im}(A)}$ 的投影阵, 则 ${\displaystyle I-P_A}$ 是到 ${\displaystyle {\mathrm{Im}}^{\perp }(A)}$ 的投影阵, 记为 ${\displaystyle P_{A^{\mathrm{T}}}}$ .

4. ${\displaystyle A,B}$ 为 ${\displaystyle n}$ 阶对称阵, ${\displaystyle C=A+B}$ 幂等, 则

   1. ${\displaystyle A,B}$ 幂等等价于 ${\displaystyle AB=0}$.
   2. ${\displaystyle A}$ 幂等, ${\displaystyle B\ge 0\Rightarrow B}$ 幂等.

5. ${\displaystyle {\mathcal{L}}_1,{\mathcal{L}}_2}$ 是 ${\displaystyle {\mathbb{R}}^n}$ 的线性子空间, 则下列命题等价:

   • ${\displaystyle {\mathcal{L}}_2\subset {\mathcal{L}}_1}$.
   • ${\displaystyle P_{{\mathcal{L}}_1}-P_{{\mathcal{L}}_2}\ge 0}$.
   • ${\displaystyle P_{{\mathcal{L}}_1}{P}_{{\mathcal{L}}_2}=P_{{\mathcal{L}}_2}}$.
   • ${\displaystyle P_{{\mathcal{L}}_2}{P}_{{\mathcal{L}}_1}=P_{{\mathcal{L}}_2}}$.

   且上面任一条件满足时 ${\displaystyle P_{{\mathcal{L}}_1}-P_{{\mathcal{L}}_2}=P_{{\mathcal{L}}_1\cap {\mathcal{L}}_2^{\perp }}}$ 是到 ${\displaystyle {\mathcal{L}}_1\cap {\mathcal{L}}_2^{\perp }}$ 的正投影阵.