Frobenius inner product

In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted ${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}}$. 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.

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

${\displaystyle \mathbf{𝐀}=\left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1m}\\ A_{21}& A_{22}& \cdots & A_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ A_{n1}& A_{n2}& \cdots & A_{nm}\\ \end{array}\right),\mathbf{𝐁}=\left(\begin{array}{cccc}{B}_{11}& B_{12}& \cdots & B_{1m}\\ B_{21}& B_{22}& \cdots & B_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ B_{n1}& B_{n2}& \cdots & B_{nm}\\ \end{array}\right)}$

the Frobenius inner product is defined as,

${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}=\sum _{i,j}\overline{A_{ij}}{B}_{ij}=\mathrm{T}\mathrm{r}\left(\overline{{\mathbf{𝐀}}^T}\mathbf{𝐁}\right)\equiv \mathrm{T}\mathrm{r}\left({\mathbf{𝐀}}^{†}\mathbf{𝐁}\right)}$

where the overline denotes the complex conjugate, and ${\displaystyle †}$ denotes Hermitian conjugate.^[1] Explicitly this sum is

${\displaystyle \begin{array}{rl}⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}=& {\bar{A}}_{11}{B}_{11}+{\bar{A}}_{12}{B}_{12}+\cdots +{\bar{A}}_{1m}{B}_{1m}\\ & +{\bar{A}}_{21}{B}_{21}+{\bar{A}}_{22}{B}_{22}+\cdots +{\bar{A}}_{2m}{B}_{2m}\\ & ⋮\\ & +{\bar{A}}_{n1}{B}_{n1}+{\bar{A}}_{n2}{B}_{n2}+\cdots +{\bar{A}}_{nm}{B}_{nm}\\ \end{array}}$

The calculation is very similar to the dot product, which in turn is an example of an inner product.^{[citation needed]}

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 "${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}(\cdot )}$"), then

${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐀})=\left(\begin{array}{c}{A}_{11}\\ A_{12}\\ ⋮\\ A_{21}\\ A_{22}\\ ⋮\\ A_{nm}\end{array}\right),\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐁})=\left(\begin{array}{c}{B}_{11}\\ B_{12}\\ ⋮\\ B_{21}\\ B_{22}\\ ⋮\\ B_{nm}\end{array}\right),}$${\displaystyle \stackrel{T}{\stackrel{‾}{\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐀})}}\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐁})=\left(\begin{array}{ccccccc}{\bar{A}}_{11}& {\bar{A}}_{12}& \cdots & {\bar{A}}_{21}& {\bar{A}}_{22}& \cdots & {\bar{A}}_{nm}\end{array}\right)\left(\begin{array}{c}{B}_{11}\\ B_{12}\\ ⋮\\ B_{21}\\ B_{22}\\ ⋮\\ B_{nm}\end{array}\right)}$

Therefore

${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}=\stackrel{T}{\stackrel{‾}{\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐀})}}\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{𝐁}).}$^{[citation needed]}

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:

${\displaystyle ⟨a\mathbf{𝐀},b\mathbf{𝐁}{⟩}_{\mathrm{F}}=\bar{a}b⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}}$

${\displaystyle ⟨\mathbf{𝐀}+\mathbf{𝐂},\mathbf{𝐁}+\mathbf{𝐃}{⟩}_{\mathrm{F}}=⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}+⟨\mathbf{𝐀},\mathbf{𝐃}{⟩}_{\mathrm{F}}+⟨\mathbf{𝐂},\mathbf{𝐁}{⟩}_{\mathrm{F}}+⟨\mathbf{𝐂},\mathbf{𝐃}{⟩}_{\mathrm{F}}}$

Also, exchanging the matrices amounts to complex conjugation:

${\displaystyle ⟨\mathbf{𝐁},\mathbf{𝐀}{⟩}_{\mathrm{F}}=\overline{⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}}}$

For the same matrix,

${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐀}{⟩}_{\mathrm{F}}\ge 0}$,^{[citation needed]}

and,

${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐀}{⟩}_{\mathrm{F}}=0⟺\mathbf{𝐀}=\mathbf{𝟎}}$.

The inner product induces the Frobenius norm

${\displaystyle \Vert \mathbf{𝐀}{\Vert }_{\mathrm{F}}=\sqrt{⟨\mathbf{𝐀},\mathbf{𝐀}{⟩}_{\mathrm{F}}}.}$^[1]

For two real-valued matrices, if

${\displaystyle \mathbf{𝐀}=\left(\begin{array}{ccc}2& 0& 6\\ 1& -1& 2\end{array}\right),\mathbf{𝐁}=\left(\begin{array}{ccc}8& -3& 2\\ 4& 1& -5\end{array}\right)}$

then

${\displaystyle \begin{array}{rl}⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}& =2\cdot 8+0\cdot (-3)+6\cdot 2+1\cdot 4+(-1)\cdot 1+2\cdot (-5)\\ & =21\end{array}}$

For two complex-valued matrices, if

${\displaystyle \mathbf{𝐀}=\left(\begin{array}{cc}1+i& -2i\\ 3& -5\end{array}\right),\mathbf{𝐁}=\left(\begin{array}{cc}-2& 3i\\ 4-3i& 6\end{array}\right)}$

then

${\displaystyle \begin{array}{rl}⟨\mathbf{𝐀},\mathbf{𝐁}{⟩}_{\mathrm{F}}& =(1-i)\cdot (-2)+2i\cdot 3i+3\cdot (4-3i)+(-5)\cdot 6\\ & =-26-7i\end{array}}$

while

${\displaystyle \begin{array}{rl}⟨\mathbf{𝐁},\mathbf{𝐀}{⟩}_{\mathrm{F}}& =(-2)\cdot (1+i)+(-3i)\cdot (-2i)+(4+3i)\cdot 3+6\cdot (-5)\\ & =-26+7i\end{array}}$

The Frobenius inner products of A with itself, and B with itself, are respectively

${\displaystyle ⟨\mathbf{𝐀},\mathbf{𝐀}{⟩}_{\mathrm{F}}=2+4+9+25=40}$${\displaystyle ⟨\mathbf{𝐁},\mathbf{𝐁}{⟩}_{\mathrm{F}}=4+9+25+36=74}$

• Hadamard product (matrices)
• Hilbert–Schmidt inner product
• Kronecker product
• Matrix analysis
• Matrix multiplication
• Matrix norm
• Tensor product of Hilbert spaces – the Frobenius inner product is the special case where the vector spaces are finite-dimensional real or complex vector spaces with the usual Euclidean inner product

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Frobenius inner product. Read more