Hadamard积及其等式性质

定义

m × n \mathit{m} \times \mathit{n} m×n矩阵 A = [ a i j ] \boldsymbol A=[\mathit{a} _{\mathit{ij} } ] A=[aij] m × n \mathit{m} \times \mathit{n} m×n矩阵 B = [ b i j ] \boldsymbol B=[\mathit{b} _{\mathit{ij} } ] B=[bij] H a d a m a r d \mathit{Hadamard} Hadamard积记作 A ∗ B \boldsymbol A\ast \boldsymbol B AB,它仍然是一个 m × n \mathit{m} \times \mathit{n} m×n的矩阵,其元素定义为两个矩阵对应元素的乘积 ( A ∗ B ) i j = a i j b i j (\boldsymbol A\ast \boldsymbol B)_{\mathit{ij} } =\mathit{a} _{\mathit{ij} }\mathit{b} _{\mathit{ij} } (AB)ij=aijbij H a d a m a r d \mathit{Hadamard} Hadamard积也称 S c h u r \mathit{Schur} Schur积或者对应元素乘积。

H a d a m a r d \mathit{Hadamard} Hadamard积满足交换律、结合律以及加法的分配率

A ∗ B = B ∗ A A ∗ ( B ∗ C ) = ( A ∗ B ) ∗ C A ∗ ( B ± C ) = A ∗ B ± A ∗ C \begin{aligned} \boldsymbol A *\boldsymbol B &=\boldsymbol B *\boldsymbol A \\ \boldsymbol A *(\boldsymbol B *\boldsymbol C) &=(\boldsymbol{A} * \boldsymbol{B}) * \boldsymbol{C} \\ \boldsymbol{A} *(\boldsymbol{B} \pm \boldsymbol{C}) &=\boldsymbol{A} * \boldsymbol{B} \pm \boldsymbol{A} * \boldsymbol{C} \end{aligned} ABA(BC)A(B±C)=BA=(AB)C=AB±AC

H a d a m a r d \mathit{Hadamard} Hadamard积的性质

  1. A , B \boldsymbol A,\boldsymbol B A,B均为 m × n \mathit{m} \times \mathit{n} m×n矩阵,则 ( A ∗ B ) T = A T ∗ B T , ( A ∗ B ) H = A H ∗ B H , ( A ∗ B ) ∗ = A ∗ ∗ B ∗ (\boldsymbol{A} * \boldsymbol{B})^{\mathrm{T}}=\boldsymbol{A}^{\mathrm{T}} * \boldsymbol{B}^{\mathrm{T}}, \quad(\boldsymbol{A} * \boldsymbol{B})^{\mathrm{H}}=\boldsymbol{A}^{\mathrm{H}} * \boldsymbol{B}^{\mathrm{H}}, \quad(\boldsymbol{A} * \boldsymbol{B})^{*}=\boldsymbol{A}^{*} * \boldsymbol{B}^{*} (AB)T=ATBT,(AB)H=AHBH,(AB)=AB
  2. 矩阵 A m × n \boldsymbol A_{\mathit{m} \times \mathit{n} } Am×n与零矩阵 O m × n \boldsymbol O_\mathit{{m} \times \mathit{n} } Om×n H a d a m a r d \mathit{Hadamard} Hadamard A ∗ O m × n = O m × n ∗ A = O m × n \boldsymbol A * \boldsymbol O_{\mathit{m} \times \mathit{n} }=\boldsymbol O_{\mathit{m} \times \mathit{n} } *\boldsymbol A=\boldsymbol O_{\mathit{m} \times \mathit{n} } AOm×n=Om×nA=Om×n
  3. c \mathit{c} c为常数,则 c ( A ∗ B ) = ( c A ) ∗ B = A ∗ ( c B ) \mathit{c} (\boldsymbol{A} * \boldsymbol{B})=(\mathit{c} \boldsymbol{A}) * \boldsymbol{B}=\boldsymbol{A} *(\mathit{c} \boldsymbol{B}) c(AB)=(cA)B=A(cB)
  4. 正定(或半正定)的矩阵 A , B \boldsymbol A,\boldsymbol B A,B H a d a m a r d \mathit{Hadamard} Hadamard A ∗ B \boldsymbol{A} * \boldsymbol{B} AB也是正定(或半正定)的。
  5. 矩阵 A m × m = [ a i j ] \boldsymbol{A}_{\mathit{m} \times \mathit{m} }=\left[\mathit{a} _{\mathit{i j} }\right] Am×m=[aij]与单位矩阵 I m \boldsymbol{I}_{\mathit{m} } Im H a d a m a r d \mathit{Hadamard} Hadamard积为 m × m \mathit{m} \times \mathit{m} m×m对角矩阵,即 A ∗ I m = I m ∗ A = d i a g ( A ) = d i a g ( a 11 , a 22 , ⋯   , a m m ) \boldsymbol{A} * \boldsymbol{I}_{\mathit{m} }=\boldsymbol{I}_{\mathit{m} } * \boldsymbol{A}=\mathit{diag}(\boldsymbol{A})=\mathit{diag}\left(\mathit{a} _{11}, \mathit{a} _{22}, \cdots, \mathit{a} _{\mathit{m m} }\right) AIm=ImA=diag(A)=diag(a11,a22,,amm)
  6. A , B , D \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{D} A,B,D m × m \mathit{m} \times \mathit{m} m×m矩阵,且 D \boldsymbol{D} D为对角矩阵,则 ( D A ) ∗ ( B D ) = D ( A ∗ B ) D (\boldsymbol{D} \boldsymbol{A}) *(\boldsymbol{B} \boldsymbol{D})=\boldsymbol{D}(\boldsymbol{A} * \boldsymbol{B}) \boldsymbol{D} (DA)(BD)=D(AB)D
  7. A , C \boldsymbol{A}, \boldsymbol{C} A,C m × m \mathit{m} \times \mathit{m} m×m矩阵,并且 B , D \boldsymbol{B}, \boldsymbol{D} B,D n × n \mathit{n} \times \mathit{n} n×n矩阵,则 ( A ⊕ B ) ∗ ( C ⊕ D ) = ( A ∗ C ) ⊕ ( B ∗ D ) (\boldsymbol{A} \oplus \boldsymbol{B}) *(\boldsymbol{C} \oplus \boldsymbol{D})=(\boldsymbol{A} * \boldsymbol{C}) \oplus(\boldsymbol{B} * \boldsymbol{D}) (AB)(CD)=(AC)(BD)
  8. A , B , C , D \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{D} A,B,C,D均为 m × n \mathit{m} \times \mathit{n} m×n矩阵,则 ( A + B ) ∗ ( C + D ) = A ∗ C + A ∗ D + B ∗ C + B ∗ D (\boldsymbol A+\boldsymbol B) *(\boldsymbol C+\boldsymbol D)=\boldsymbol A *\boldsymbol C+\boldsymbol A *\boldsymbol D+\boldsymbol B *\boldsymbol C+\boldsymbol B *\boldsymbol D (A+B)(C+D)=AC+AD+BC+BD
  9. A , B , C \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} A,B,C m × n \mathit{m} \times \mathit{n} m×n矩阵,则 t r ( A T ( B ∗ C ) ) = t r ( ( A T ∗ B T ) C ) \mathit{tr} \left(\boldsymbol{A}^{\mathrm{T}}(\boldsymbol{B} * \boldsymbol{C})\right)=\mathit{tr} \left(\left(\boldsymbol{A}^{\mathrm{T}} * \boldsymbol{B}^{\mathrm{T}}\right) \boldsymbol{C}\right) tr(AT(BC))=tr((ATBT)C)

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