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矩阵基本性质

矩阵\(A\)的第\(i\)⾏第\(j\)列的元素为\(A_{\text{ij}}\)。我们⽤\(I_{N}\)或(\(I\))表⽰\(N \times N\)的单位矩阵。

1.矩阵的加减法

(1)\({C}={A \pm B}\),对应元素相加减

(2)矩阵加减法满足的运算法则

a.交换律:\({A + B}={B + A}\)

b.结合律:\(\left( {A + B} \right) + {B}{= A+}({B + C})\)

c.\({A + 0}={A}\)

d.\(A - A= 0\)

2.矩阵的数乘

(1)\(\bf{B}=\lambda\bf{A}\),各元素均乘以常数

(2)矩阵数乘满足的运算法则

a.数对矩阵的分配律:\(\lambda\left( \bf{A + B}\right)\bf=\lambda\bf{A +}\lambda\bf{B}\)

b.矩阵对数的分配律:\(\left( \lambda\bf{+}\mu\right)\bf{A}\bf=\lambda\bf{A +}\mu\bf{A}\)

c.结合律:\(\left( \text{λμ}\right)\bf{A}\bf=\lambda\bf{(}\mu\bf{A}\bf{)}\)

d.\(\bf{ 0 \bullet A}\bf{= 0}\)

3.矩阵的乘法

(1)\(\bf{C}\bf=\bf{A}_{\bf{l \times n}}\bf{B}_{\bf{n\times m}}\),左行右列对应元素相乘后求和为C的第\(\bf{l}\)行第\(\bf{m}\)列的元素

(2)矩阵乘法满足的运算法则

a.对于一般矩阵不满足交换律,只有两个方正满足且有\(\bf{\text{AI}}\bf=\bf{IA= A}\)

b.分配律:\(\bf{A}\left( \bf{B + C} \right)\bf{=AB}\bf{+}\bf{\text{AC}}\)

c.结合律:\(\bf{\ }\left( \bf{\text{AB}}\right)\bf{C}\bf=\bf{A}\bf{(}\bf{\text{BC}}\bf{)}\)

d.数乘结合律:\(\lambda\left( \bf{\text{AB}}\right)\bf=\bf{A}\bf{(}\lambda\bf{B}\bf{)}\)

4.矩阵的转置

\(\bf{A}^{\bf{T}}\),\({\bf{(}\bf{A}^{\bf{T}}\bf{)}}_{\text{ij}} = A_{\text{ji}}\)

(1)矩阵的幂:\(A^{1} = A\),\(\ A^{2} = \text{AA}\),…,\(\ A^{k + 1} = A(A^{k})\)

(2)矩阵乘法满足的运算法则

a.\({\bf{(}\bf{A}^{\bf{T}}\bf{)}}^{\bf{T}}\bf{= A}\)

b.\(\bf{\ }{\bf{(A + B}\bf{)}}^{\bf{T}} =\bf{A}^{\bf{T}}\bf{+}\bf{B}^{\bf{T}}\)

c.\(\bf{\ }{\bf{(}\lambda\bf{A}\bf{)}}^{\bf{T}} =\lambda(\bf{A}^{\bf{T}}\bf{)}\)

d.\(\bf{\ }{\bf{(AB}\bf{)}}^{\bf{T}} =\bf{A}^{\bf{T}}\bf{B}^{\bf{T}}\)

5.对称矩阵

\(\bf{A}^{\bf{T}}\bf{= A}\)\(a_{\text{ij}} =a_{\text{ji}}\)

反对称矩阵:\(\bf{A}^{\bf{T}}\bf{= - A}\)\(a_{\text{ij}} = {-a}_{\text{ji}}\)

(1)设\(\bf{A}\bf{,}\bf{B}\)为(反)对称矩阵,则\(\bf{A \pm B}\)仍是(反)对称矩阵。

(2)设\(\bf{A}\bf{,}\bf{B}\)为对称矩阵,则\(\bf{\text{AB}}\)\(\bf{\text{BA}}\)仍是对称矩阵的充要条件\(\bf{\text{AB}}\)=\(\bf{\text{BA}}\)

(3)设\(\bf{A}\)为(反)对称矩阵,则\(\bf{A}^{\bf{T}}\)\(\lambda\bf{A}\)也是(反)对称矩阵。

(4)对任意矩阵\(\bf{A}\),则\(\bf{H} \equiv \frac{1}{2}\left(\bf{A} + \bf{A}^{\bf{T}} \right),\bf{S} \equiv \frac{1}{2}\left(\bf{A} + \bf{A}^{\bf{T}}\right)\)分别是对称矩阵和反对称矩阵且\(\bf{A} = \bf{H} + \bf{S}\).

(5)\({\bf{(}\bf{A}^{\bf{T}}\bf{)}}^{\bf{T}} = \bf{A}\)

6. Hermite矩阵:

\(\bf{A}^{\bf{H}}\bf{= A}\),即\(a_{\text{ij}}= \overset{\overline{}}{a_{\text{ji}}}\)

反Hermite矩阵,\(\bf{A}^{\bf{H}}\bf{= - A}\)\(a_{\text{ij}} =\overset{\overline{}}{a_{\text{ji}}}\)

a.\(\bf{A}^{\bf{H}}\bf={\bf{(}\overset{\overline{}}{A}\bf{)}}^{\bf{T}}\)

b.\(\bf{\ }{\bf{(A + B}\bf{)}}^{\bf{H}} =\bf{A}^{\bf{H}}\bf{+}\bf{B}^{\bf{H}}\)

c.\(\bf{\ }{\bf{(}\lambda\bf{A}\bf{)}}^{\bf{H}} =\overset{\overline{}}{\lambda}(\bf{A}^{\bf{H}}\bf{)}\)

d.\(\bf{\ }{\bf{(AB}\bf{)}}^{\bf{H}} =\bf{A}^{\bf{H}}\bf{B}^{\bf{H}}\)

  1. \({\bf{(}\bf{A}^{\bf{T}}\bf{)}}^{\bf{T}} = \bf{A}\)

f.\(\bf{\ }{\bf{(}\bf{A}^{\bf{H}}\bf{)}}^{\bf{- 1}} ={\bf{(}\bf{A}^{\bf{-1}}\bf{)}}^{\bf{H}}\)(当\(\bf{A}\)矩阵可逆时)

7.正交矩阵

\(\bf{A}^{\bf{T}}\bf{A = A}\bf{A}^{\bf{T}}\bf{=E}\),,则\(\bf{A,(B) \in}\bf{E}^{\bf{n \times n}}\)是正交矩阵

(1)\(\bf{A}^{- \bf{1}} =\bf{A}^{\bf{T}}\bf{\in}\bf{E}^{\bf{n \times n}}\)

(2)\(\det\bf{A}\bf{= \pm}1\)

(3)\(\bf{\text{AB}}\),\(\bf{\ BA \in}\bf{E}^{\bf{n \times n}}\)

8.酉矩阵

\(\bf{A}^{\bf{H}}\bf{A = A}\bf{A}^{\bf{H}}\bf{=E}\),则\(\bf{A,(B) \in}\bf{U}^{\bf{n \times n}}\)是酉矩阵

(1)\(\bf{A}^{- \bf{1}} =\bf{A}^{\bf{H}}\bf{\in}\bf{U}^{\bf{n \times n}}\)

(2)\(\left| \det\bf{A} \right|\bf=1\)

(3)\(\bf{\text{AB}}\),\(\bf{\ BA \in}\bf{U}^{\bf{n \times n}}\)

(4)\(\bf{A}^{\bf{T}}\bf{\in}\bf{U}^{\bf{n \times n}}\)

9.正规矩阵

\(\bf{A}^{\bf{H}}\bf{A =A}\bf{A}^{\bf{H}}\),则\(\bf{A}\)是正规矩阵;

\(\bf{A}^{\bf{T}}\bf{A =A}\bf{A}^{\bf{T}}\),则\(\bf{A}\)是实正规矩阵

10.矩阵的迹和行列式

(1)\(\text{tr}\left( \bf{A} \right) = \sum_{\bf{i} =\bf{1}}^{\bf{n}}\bf{a}_{\bf{\text{ii}}} = \sum_{\bf{i} =\bf{1}}^{\bf{n}}\bf{\lambda}_{\bf{i}}\)为矩阵\(\bf{A}\)的迹;\(\left|\bf{A} \right|\)\(det(\bf{A})\)为行列式

(2)\(\text{tr}\left( \bf{\text{AB}} \right) = \text{tr}\left(\bf{\text{BA}} \right)\);注:矩阵乘法不满足交换律

(3)\(\text{tr}\left( \bf{\text{ABC}} \right) = \text{tr}\left(\bf{\text{CBA}} \right) = \text{tr}\left( \bf{\text{BCA}} \right)\)

(4)\(A =\bf{\text{UB}}\bf{U}^{\bf{\dagger}}\),\(\bf{\text{\U}}\)为酉矩阵,则\(\text{tr}\left( \bf{A} \right) = \text{tr}\left(\bf{B} \right)\)

(5)\(\left| \bf{I}_{\bf{M}}\bf{+ A}\bf{B}^{\bf{T}}\right|\bf=\left|\bf{I}_{\bf{N}}\bf{+}\bf{A}^{\bf{T}}\bf{B} \right|\)

(6)\(\left| \bf{I}_{\bf{M}}\bf{+ a}\bf{b}^{\bf{T}}\right|\bf=\left|\bf{I}_{\bf{N}}\bf{+}\bf{a}^{\bf{T}}\bf{b} \right|\)

(7)\(\left| \bf{A}^{\bf{T}} \right|\bf=\left| \bf{A}\right|\)

(8)\(\left| \lambda\bf{A} \right|\bf=\lambda^{n}\left| \bf{A}\right|\)

(9)\(\left| \bf{\text{AB}} \right| = \left| \bf{A} \right|\left|\bf{B} \right|\)

(10)\(det(I + \bf{\text{AB}}) = det(I + \bf{\text{BA}})\)

(11)\(\left| \bf{A} \right| = \prod_{\bf{i} =\bf{1}}^{\bf{M}}\bf{\lambda}_{\bf{i}}\)

(12)\(\bf{C} = \log\left\lbrack \det\left( \bf{I}_{\bf{M}} +\bf{\text{HQ}}\bf{H}^{*} \right) \right\rbrack\),\(\ \bf{Q} =\frac{\bf{\rho}}{\bf{N}}\bf{I}_{\bf{N}}\),则\(\bf{C} =\sum_{i = 1}^{m}{\log}{\left(1+\frac{\rho}N\lambda_{i}\right)}\)其中\(\lambda_{i}\)\(\bf{H}\bf{H}^{*}\)奇异分解值的特征值

11.矩阵的伴随矩阵\(\bf{A}^{\bf{*}}\)

(1)设\(\bf{A} = \left\{ \bf{a}_{\bf{\text{ij}}}\right\}\)由行列式\(\left| \bf{A}\right|\)的代数余子式\(\bf{A}_{\bf{\text{ij}}}\)所构成的矩阵

(2)\(\bf{A}\bf{A}^{*} = \bf{A}^{*}\bf{A} = \left| A\right|\bf{I}\)

12.矩阵的逆(逆矩阵是唯一的)

(1)A的逆矩阵记作\(\bf{A}^{- \bf{1}}\),\(\ \bf{A}\bf{A}^{-\bf{1}} = \bf{A}^{- \bf{1}}\bf{A} = \bf{I}\);

(2)\(\left| \bf{A} \right| \neq0\)\(\bf{A}\)为非奇矩阵)时,\(\bf{A}^{- \bf{1}} =\frac{\bf{1}}{\left| \bf{A} \right|}\bf{A}^{*}\)

(3)\(\left| \bf{A} \right| \neq 0\)\(\lambda \neq0\),则\({\bf{(}\lambda\bf{A}\bf{)}}^{\bf{- 1}} =\frac{1}{\lambda}\bf{A}^{\bf{- 1}}\)

(4)由\(\bf{A}{\bf{B}\bf{B}^{- \bf{1}}\bf{A}}^{-\bf{1}} = \bf{I}\),得\({(\bf{\text{AB}})}^{- \bf{1}} =\bf{A}^{\bf{- 1}}\bf{B}^{\bf{- 1}}\)

(5)\({\bf{(}\bf{A}^{\bf{T}}\bf{)}}^{\bf{- 1}} ={\bf{(}\bf{A}^{\bf{- 1}}\bf{)}}^{\bf{T}}\)

(6)若\(\left| \bf{A} \right| \neq 0,\left| \bf{A}^{\bf{- 1}}\right|\bf=\frac{\bf{1}}{\left| \bf{A} \right|}\)

(7)若\(\bf{A}\)是非奇上(下)三角矩阵,则\(\bf{A}^{\bf{-1}}\)也上(下)三角矩阵

(8)\(\bf{A}^{- k} = {\bf{(}\bf{A}^{\bf{- 1}}\bf{)}}^{k}\)

(9)\({\bf{(}\bf{P}^{\bf{-1}}\bf{+}\bf{B}^{\bf{T}}\bf{R}^{\bf{-1}}\bf{B}\bf{)}}^{\bf{-1}}\bf{B}^{\bf{T}}\bf{R}^{\bf{- 1}}\bf{=P}{\bf{B}^{\bf{T}}\bf{(}\bf{\text{BPB}}^{\bf{T}}\bf{+}\bf{R}\bf{)}}^{\bf{-1}}\)

(10)\({\bf{(}\bf{I}\bf{+}\bf{\text{AB}}\bf{)}}^{\bf{-1}}\bf{A =B}{\bf{(}\bf{I}\bf{+}\bf{\text{BA}}\bf{)}}^{\bf{- 1}}\)

(11)Woodbury恒等式 \({\bf{(}\bf{A}\bf{+}\bf{B}\bf{D}^{\bf{-1}}\bf{C}\bf{)}}^{\bf{- 1}}\bf=\bf{A}^{\bf{-1}}\bf{-}\bf{A}^{\bf{- 1}}\bf{B}{\bf{(D+}\bf{C}\bf{A}^{\bf{- 1}}\bf{B}\bf{)}}^{\bf{-1}}\bf{C}\bf{A}^{\bf{- 1}}\)

(12)\(\bf{A}^{\bf{- 1}}\bf{= U} \land^{-1}\bf{U}^{\bf{H}}\)

12.对角矩阵

矩阵\(\bf{A}\)为对称矩阵,\(\bf{Q}\)正交矩阵,则\(\bf{Q}^{\bf{-1}}\bf{AQ =diag}\bf{(}\bf{\lambda}_{\bf{1}}\bf{\cdots,}\bf{\lambda}_{\bf{n}}\bf{)}\)为对角矩阵或\(\bf{U}^{\bf{-1}}\bf{AU =}\bf{U}^{\bf{H}}\bf{AU = diag}\left(\bf{\lambda}_{\bf{1}}\bf{\cdots,}\bf{\lambda}_{\bf{n}}\right)\bf= \land\),则\(\bf{A = U} \land\bf{U}^{\bf{H}}\bf=\sum_{\bf{i} =\bf{1}}^{\bf{n}}{\bf{\lambda}_{\bf{i}}u_{i}u_{i}^{T}}\);\({\bf{A}^{\bf{- 1}}\bf{= U} \land^{-1}\bf{U}^{\bf{H}}\bf=\sum_{\bf{i} =\bf{1}}^{\bf{n}}{\frac{1}\bf{\lambda}_{\bf{i}}}u_{i}u_{i}^{T}}\)

13.矩阵的导数

(1)\(\frac{\partial}{\partial x}\left( \bf{\text{AB}} \right) =\frac{\partial\bf{A}}{\partial x}\bf{B} +\bf{A}\frac{\partial\bf{B}}{\partial x}\)

(2)\(\frac{\partial}{\partial x}\left( \bf{A}^{\bf{- 1}} \right) = -\bf{A}^{\bf{- 1}}\frac{\partial\bf{A}}{\partial x}\bf{A}^{\bf{- 1}}​\)

(3)\(\frac{\partial}{\partial x}\ln\left| \bf{A} \right| = tr\left(\bf{A}^{\bf{- 1}}\frac{\partial\bf{A}}{\partial x} \right)\)

(4)\(\frac{\partial}{\partial A_{\text{ij}}}\text{tr}\bf{(AB)} =B_{\text{ij}}\)

(5)\(\frac{\partial}{\partial\bf{A}}\text{tr}\bf{(AB)} =\bf{B}^{T}\)

(6)\(\frac{\partial}{\partial\bf{A}}\text{tr}\bf{(}\bf{A}^{T}\bf{B)}= \bf{B}\)

(7)\(\frac{\partial}{\partial\bf{A}}\text{tr}\bf{(A)} = \bf{I}\)

(8)\(\frac{\partial}{\partial\bf{A}}\text{tr}\bf{(AB}\bf{A}^{T}\bf{)}= \bf{A(B +}\bf{B}^{T}\bf{)}\)

(9)\(\frac{\partial}{\partial\bf{A}}\ln\left| \bf{A} \right| = \left(\bf{A}^{\bf{- 1}} \right)^{T}\)