Binary operation, takes two matrices and returns a scalar
In mathematics , the Frobenius inner product is a binary operation that takes two matrices and returns a scalar . It is often denoted ⟨ A , B ⟩ F {\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }} . The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension - same number of rows and columns, but are not restricted to be square matrices .
Definition [ edit ] Given two complex number -valued n ×m matrices A and B , written explicitly as
A = ( A 11 A 12 ⋯ A 1 m A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n m ) , B = ( B 11 B 12 ⋯ B 1 m B 21 B 22 ⋯ B 2 m ⋮ ⋮ ⋱ ⋮ B n 1 B n 2 ⋯ B n m ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&\cdots &B_{1m}\\B_{21}&B_{22}&\cdots &B_{2m}\\\vdots &\vdots &\ddots &\vdots \\B_{n1}&B_{n2}&\cdots &B_{nm}\\\end{pmatrix}}} the Frobenius inner product is defined as,
⟨ A , B ⟩ F = ∑ i , j A i j ¯ B i j = T r ( A T ¯ B ) ≡ T r ( A † B ) {\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=\sum _{i,j}{\overline {A_{ij}}}B_{ij}\,=\mathrm {Tr} \left({\overline {\mathbf {A} ^{T}}}\mathbf {B} \right)\equiv \mathrm {Tr} \left(\mathbf {A} ^{\!\dagger }\mathbf {B} \right)} where the overline denotes the complex conjugate , and † {\displaystyle \dagger } denotes Hermitian conjugate .[1] Explicitly this sum is
⟨ A , B ⟩ F = A ¯ 11 B 11 + A ¯ 12 B 12 + ⋯ + A ¯ 1 m B 1 m + A ¯ 21 B 21 + A ¯ 22 B 22 + ⋯ + A ¯ 2 m B 2 m ⋮ + A ¯ n 1 B n 1 + A ¯ n 2 B n 2 + ⋯ + A ¯ n m B n m {\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=&{\overline {A}}_{11}B_{11}+{\overline {A}}_{12}B_{12}+\cdots +{\overline {A}}_{1m}B_{1m}\\&+{\overline {A}}_{21}B_{21}+{\overline {A}}_{22}B_{22}+\cdots +{\overline {A}}_{2m}B_{2m}\\&\vdots \\&+{\overline {A}}_{n1}B_{n1}+{\overline {A}}_{n2}B_{n2}+\cdots +{\overline {A}}_{nm}B_{nm}\\\end{aligned}}} The calculation is very similar to the dot product , which in turn is an example of an inner product.[citation needed ]
Relation to other products [ edit ] If A and B are each real -valued matrices, the Frobenius inner product is the sum of the entries of the Hadamard product . If the matrices are vectorised (i.e., converted into column vectors, denoted by " v e c ( ⋅ ) {\displaystyle \mathrm {vec} (\cdot )} "), then
v e c ( A ) = ( A 11 A 12 ⋮ A 21 A 22 ⋮ A n m ) , v e c ( B ) = ( B 11 B 12 ⋮ B 21 B 22 ⋮ B n m ) , {\displaystyle \mathrm {vec} (\mathbf {A} )={\begin{pmatrix}A_{11}\\A_{12}\\\vdots \\A_{21}\\A_{22}\\\vdots \\A_{nm}\end{pmatrix}},\quad \mathrm {vec} (\mathbf {B} )={\begin{pmatrix}B_{11}\\B_{12}\\\vdots \\B_{21}\\B_{22}\\\vdots \\B_{nm}\end{pmatrix}}\,,} v e c ( A ) ¯ T v e c ( B ) = ( A ¯ 11 A ¯ 12 ⋯ A ¯ 21 A ¯ 22 ⋯ A ¯ n m ) ( B 11 B 12 ⋮ B 21 B 22 ⋮ B n m ) {\displaystyle \quad {\overline {\mathrm {vec} (\mathbf {A} )}}^{T}\mathrm {vec} (\mathbf {B} )={\begin{pmatrix}{\overline {A}}_{11}&{\overline {A}}_{12}&\cdots &{\overline {A}}_{21}&{\overline {A}}_{22}&\cdots &{\overline {A}}_{nm}\end{pmatrix}}{\begin{pmatrix}B_{11}\\B_{12}\\\vdots \\B_{21}\\B_{22}\\\vdots \\B_{nm}\end{pmatrix}}} Therefore
⟨ A , B ⟩ F = v e c ( A ) ¯ T v e c ( B ) . {\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }={\overline {\mathrm {vec} (\mathbf {A} )}}^{T}\mathrm {vec} (\mathbf {B} )\,.} [citation needed ] Properties [ edit ] Like any inner product, it is a sesquilinear form , for four complex-valued matrices A , B , C , D , and two complex numbers a and b :
⟨ a A , b B ⟩ F = a ¯ b ⟨ A , B ⟩ F {\displaystyle \langle a\mathbf {A} ,b\mathbf {B} \rangle _{\mathrm {F} }={\overline {a}}b\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }} ⟨ A + C , B + D ⟩ F = ⟨ A , B ⟩ F + ⟨ A , D ⟩ F + ⟨ C , B ⟩ F + ⟨ C , D ⟩ F {\displaystyle \langle \mathbf {A} +\mathbf {C} ,\mathbf {B} +\mathbf {D} \rangle _{\mathrm {F} }=\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }+\langle \mathbf {A} ,\mathbf {D} \rangle _{\mathrm {F} }+\langle \mathbf {C} ,\mathbf {B} \rangle _{\mathrm {F} }+\langle \mathbf {C} ,\mathbf {D} \rangle _{\mathrm {F} }} Also, exchanging the matrices amounts to complex conjugation:
⟨ B , A ⟩ F = ⟨ A , B ⟩ F ¯ {\displaystyle \langle \mathbf {B} ,\mathbf {A} \rangle _{\mathrm {F} }={\overline {\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }}}} For the same matrix,
⟨ A , A ⟩ F ≥ 0 {\displaystyle \langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }\geq 0} ,[citation needed ] and,
⟨ A , A ⟩ F = 0 ⟺ A = 0 {\displaystyle \langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }=0\Longleftrightarrow \mathbf {A} =\mathbf {0} } . Frobenius norm [ edit ] The inner product induces the Frobenius norm
‖ A ‖ F = ⟨ A , A ⟩ F . {\displaystyle \|\mathbf {A} \|_{\mathrm {F} }={\sqrt {\langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }}}\,.} [1] Examples [ edit ] Real-valued matrices [ edit ] For two real-valued matrices, if
A = ( 2 0 6 1 − 1 2 ) , B = ( 8 − 3 2 4 1 − 5 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}2&0&6\\1&-1&2\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}8&-3&2\\4&1&-5\end{pmatrix}}} then
⟨ A , B ⟩ F = 2 ⋅ 8 + 0 ⋅ ( − 3 ) + 6 ⋅ 2 + 1 ⋅ 4 + ( − 1 ) ⋅ 1 + 2 ⋅ ( − 5 ) = 21 {\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }&=2\cdot 8+0\cdot (-3)+6\cdot 2+1\cdot 4+(-1)\cdot 1+2\cdot (-5)\\&=21\end{aligned}}} Complex-valued matrices [ edit ] For two complex-valued matrices, if
A = ( 1 + i − 2 i 3 − 5 ) , B = ( − 2 3 i 4 − 3 i 6 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}1+i&-2i\\3&-5\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}-2&3i\\4-3i&6\end{pmatrix}}} then
⟨ A , B ⟩ F = ( 1 − i ) ⋅ ( − 2 ) + 2 i ⋅ 3 i + 3 ⋅ ( 4 − 3 i ) + ( − 5 ) ⋅ 6 = − 26 − 7 i {\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }&=(1-i)\cdot (-2)+2i\cdot 3i+3\cdot (4-3i)+(-5)\cdot 6\\&=-26-7i\end{aligned}}} while
⟨ B , A ⟩ F = ( − 2 ) ⋅ ( 1 + i ) + ( − 3 i ) ⋅ ( − 2 i ) + ( 4 + 3 i ) ⋅ 3 + 6 ⋅ ( − 5 ) = − 26 + 7 i {\displaystyle {\begin{aligned}\langle \mathbf {B} ,\mathbf {A} \rangle _{\mathrm {F} }&=(-2)\cdot (1+i)+(-3i)\cdot (-2i)+(4+3i)\cdot 3+6\cdot (-5)\\&=-26+7i\end{aligned}}} The Frobenius inner products of A with itself, and B with itself, are respectively
⟨ A , A ⟩ F = 2 + 4 + 9 + 25 = 40 {\displaystyle \langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }=2+4+9+25=40} ⟨ B , B ⟩ F = 4 + 9 + 25 + 36 = 74 {\displaystyle \qquad \langle \mathbf {B} ,\mathbf {B} \rangle _{\mathrm {F} }=4+9+25+36=74} See also [ edit ] References [ edit ]
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