矩阵的向量化及内积
定义1. 设矩阵 A = ( a i j ) ∈ R m × n \boldsymbol{A} = (a_{ij})\in R^{m\times n} A=(aij)∈Rm×n , 把矩阵 A \boldsymbol{A} A 的元素按行的顺序排列成一个列向量：
v e c A = ( a 11 , a 12 , ⋅ ⋅ ⋅ , a 1 n , a 21 , a 22 , ⋅ ⋅ ⋅ , a 2 n , ⋅ ⋅ ⋅ , a m 1 , a m 2 , ⋅ ⋅ ⋅ , a m n ) T vec\boldsymbol{A} = (a_{11},a_{12},\cdot\cdot\cdot,a_{1n},a_{21},a_{22},\cdot\cdot\cdot,a_{2n},\cdot\cdot\cdot,a_{m1},a_{m2},\cdot\cdot\cdot,a_{mn})^T vecA=(a11,a12,⋅⋅⋅,a1n,a21,a22,⋅⋅⋅,a2n,⋅⋅⋅,am1,am2,⋅⋅⋅,amn)T
则称向量 v e c A vec\boldsymbol{A} vecA 为矩阵 A \boldsymbol{A} A 按行展开的列向量。
定义2. 设矩阵 A = ( a i j ) ∈ R m × n \boldsymbol{A} = (a_{ij})\in R^{m\times n} A=(aij)∈Rm×n , 把矩阵 A \boldsymbol{A} A 的元素按列的顺序排列成一个列向量：
v e c A = ( a 11 , a 21 , ⋅ ⋅ ⋅ , a n 1 , a 12 , a 22 , ⋅ ⋅ ⋅ , a n 2 , ⋅ ⋅ ⋅ , a 1 m , a 2 m , ⋅ ⋅ ⋅ , a n m ) T vec\boldsymbol{A} = (a_{11},a_{21},\cdot\cdot\cdot,a_{n1},a_{12},a_{22},\cdot\cdot\cdot,a_{n2},\cdot\cdot\cdot,a_{1m},a_{2m},\cdot\cdot\cdot,a_{nm})^T vecA=(a11,a21,⋅⋅⋅,an1,a12,a22,⋅⋅⋅,an2,⋅⋅⋅,a1m,a2m,⋅⋅⋅,anm)T
则称向量 v e c A vec\boldsymbol{A} vecA 为矩阵 A \boldsymbol{A} A 按列展开的列向量。
定义3. 设 A , B ∈ R n × n \boldsymbol{A,B}\in R^{n\times n} A,B∈Rn×n，称
A ⋅ B = < A , B > = T r ( A T B ) = ∑ i = 1 n ∑ j = 1 n a i j b i j = ( v e c A ) T v e c B \boldsymbol{A\cdot B} = <\boldsymbol{A,B}>=Tr(\boldsymbol{A^TB})=\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_{ij}b_{ij}=(vec\boldsymbol{A})^Tvec\boldsymbol{B} A⋅B=<A,B>=Tr(ATB)=i=1∑nj=1∑naijbij=(vecA)TvecB
为矩阵 A , B \boldsymbol{A,B} A,B的内积。其中， T r a c e ( A ) Trace(\boldsymbol{A}) Trace(A) 为矩阵 A \boldsymbol{A} A 的迹，简记为 T r ( A ) Tr(\boldsymbol{A}) Tr(A)。
以上内容编辑：崔健棣