Frobenius inner product

Given two complex number-valued n×m matrices A and B, written explicitly as

${\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&\cdots &B_{1m}\\B_{21}&B_{22}&\cdots &B_{2m}\\\vdots &\vdots &\ddots &\vdots \\B_{n1}&B_{n2}&\cdots &B_{nm}\\\end{pmatrix}}}$

the Frobenius inner product is defined by the following summation Σ of matrix elements,

${\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=\sum _{i,j}{\overline {A_{ij}}}B_{ij}\,=\mathrm {Tr} \left({\overline {\mathbf {A} ^{T}}}\mathbf {B} \right)\equiv \mathrm {Tr} \left(\mathbf {A} ^{\!\dagger }\mathbf {B} \right)}$

where the overline denotes the complex conjugate, and ${\displaystyle \dagger }$ denotes Hermitian conjugate. Explicitly this sum is

${\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=&{\overline {A}}_{11}B_{11}+{\overline {A}}_{12}B_{12}+\cdots +{\overline {A}}_{1m}B_{1m}\\&+{\overline {A}}_{21}B_{21}+{\overline {A}}_{22}B_{22}+\cdots +{\overline {A}}_{2m}B_{2m}\\&\vdots \\&+{\overline {A}}_{n1}B_{n1}+{\overline {A}}_{n2}B_{n2}+\cdots +{\overline {A}}_{nm}B_{nm}\\\end{aligned}}}$

The calculation is very similar to the dot product, which in turn is an example of an inner product.

The inner product induces the Frobenius norm

${\displaystyle \|\mathbf {A} \|_{\mathrm {F} }={\sqrt {\langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }}}\,.}$

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 (denoted by "vec", converted into column vectors) as follows,

${\displaystyle \mathrm {vec} (\mathbf {A} )={\begin{pmatrix}A_{11}\\A_{12}\\\vdots \\A_{21}\\A_{22}\\\vdots \\A_{nm}\end{pmatrix}},\quad \mathrm {vec} (\mathbf {B} )={\begin{pmatrix}B_{11}\\B_{12}\\\vdots \\B_{21}\\B_{22}\\\vdots \\B_{nm}\end{pmatrix}}\,,}$

the matrix product

${\displaystyle {\overline {\mathrm {vec} (\mathbf {A} )}}^{T}\mathrm {vec} (\mathbf {B} )={\begin{pmatrix}{\overline {A}}_{11}&{\overline {A}}_{12}&\cdots &{\overline {A}}_{21}&{\overline {A}}_{22}&\cdots &{\overline {A}}_{nm}\end{pmatrix}}{\begin{pmatrix}B_{11}\\B_{12}\\\vdots \\B_{21}\\B_{22}\\\vdots \\B_{nm}\end{pmatrix}}}$

reproduces the definition, therefore

${\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }={\overline {\mathrm {vec} (\mathbf {A} )}}^{T}\mathrm {vec} (\mathbf {B} )\,.}$

• Hadamard product (matrices)
• Hilbert–Schmidt inner product
• Kronecker product
• Matrix analysis
• Matrix multiplication
• Matrix norm