Frobenius dot product

In linear algebra, the Frobenius scalar product is a scalar product on the vector space of the real or complex matrices . It is calculated by component-wise multiplication of the entries of two matrices and subsequent summation over all these products. In the complex case, one element is always complex conjugated . The Frobenius scalar product can also be calculated as a trace of the matrix product of the two matrices, one of the matrices being transposed or adjoint .

With the Frobenius scalar product, the matrix space becomes a scalar product space . The deduced from the scalar Frobenius norm is Frobenius norm . A generalization of the Frobenius scalar product to infinitely dimensional vector spaces is the Hilbert-Schmidt scalar product . The Frobenius scalar product is used, among other things, in continuum mechanics for the tensorial description of the deformation of vector fields . It is named after the German mathematician Ferdinand Georg Frobenius .

definition

The Frobenius scalar product of two, not necessarily quadratic , real matrices and is defined as ${\displaystyle A=(a_{ij})\in \mathbb {R} ^{m\times n}}$ ${\displaystyle B=(b_{ij})\in \mathbb {R} ^{m\times n}}$

${\displaystyle \langle A,B\rangle _{F}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}}$.

The Frobenius scalar product thus arises from the component-wise multiplication of the entries of the two output matrices and subsequent summation over all these products. It therefore corresponds to the standard scalar product if one regards the matrices as -dimensional vectors. ${\displaystyle m\cdot n}$

Correspondingly, the Frobenius scalar product of two complex matrices and is through ${\displaystyle A\in \mathbb {C} ^{m\times n}}$ ${\displaystyle B\in \mathbb {C} ^{m\times n}}$

${\displaystyle \langle A,B\rangle _{F}=\sum _{i=1}^{m}\sum _{j=1}^{n}{\bar {a}}_{ij}b_{ij}}$

where the overline represents the conjugate of a complex number . As an alternative definition, the second component can be complex conjugated instead of the first.

In physics, the Frobenius scalar product of two matrices and also is noted. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\,\colon \,B}$

example

The Frobenius scalar product of the two real (2 × 2) matrices

${\displaystyle A={\begin{pmatrix}3&2\\0&1\end{pmatrix}}}$   and   ${\displaystyle B={\begin{pmatrix}1&2\\3&1\end{pmatrix}}}$

is given by

${\displaystyle \langle A,B\rangle _{F}=3\cdot 1+2\cdot 2+0\cdot 3+1\cdot 1=8}$.

The Frobenius dot product of the two complex (2 × 2) matrices

${\displaystyle A={\begin{pmatrix}3&2i\\0&i\end{pmatrix}}}$   and   ${\displaystyle B={\begin{pmatrix}i&2\\3&-i\end{pmatrix}}}$

is according to this

${\displaystyle \langle A,B\rangle _{F}=3\cdot i+(-2i)\cdot 2+0\cdot 3+(-i)\cdot (-i)=-(i+1)}$.

properties

Dot product axioms

The following axioms of a complex scalar product are listed for the first variant, for the second variant they apply analogously by shifting the conjugation. The real case is obtained from the complex case by omitting the conjugation. The complex Frobenius scalar product is sesquilinear , that is, semilinear in the first argument, that is

${\displaystyle \langle A+B,C\rangle _{F}=\langle A,C\rangle _{F}+\langle B,C\rangle _{F}}$   and   ${\displaystyle \langle cA,B\rangle _{F}={\bar {c}}\langle A,B\rangle _{F}}$

as well as linear in the second argument, i.e.

${\displaystyle \langle A,B+C\rangle _{F}=\langle A,B\rangle _{F}+\langle A,C\rangle _{F}}$   and   ${\displaystyle \langle A,cB\rangle _{F}=c\langle A,B\rangle _{F}}$

Further it is Hermitian , that is

${\displaystyle \langle A,B\rangle _{F}={\overline {\langle B,A\rangle _{F}}}}$,

and definitely positive , so

${\displaystyle \langle A,A\rangle _{F}\geq 0}$   and   . ${\displaystyle \langle A,A\rangle _{F}=0\Leftrightarrow A=0}$

These properties follow directly from the commutative and distributive laws of addition and multiplication, as well as the positive definiteness of the complex absolute value function . In the second complex variant, the Frobenius scalar product is linear in the first and semilinear in the second argument. In the special case of two single- row or single-column matrices, the Frobenius scalar product corresponds to the standard scalar product of the two row or column vectors. With the Frobenius scalar product, the matrix space becomes a scalar product space , even a Hilbert space . ${\displaystyle |z|^{2}={\bar {z}}z}$

Representation as a trace

The real Frobenius scalar product has the following representation as a trace

${\displaystyle \langle A,B\rangle _{F}=\operatorname {spur} (A^{T}B)=\operatorname {spur} (BA^{T})}$,

where is the transposed matrix of . The complex Frobenius scalar product has the representation accordingly ${\displaystyle A^{T}}$ ${\displaystyle A}$

${\displaystyle \langle A,B\rangle _{F}=\operatorname {spur} (A^{H}B)=\operatorname {spur} (BA^{H})}$,

where is the adjoint matrix of . ${\displaystyle A^{H}}$ ${\displaystyle A}$

Displacement property

The real Frobenius scalar product has the following displacement property for all and : ${\displaystyle A\in \mathbb {R} ^{l\times m},B\in \mathbb {R} ^{m\times n}}$ ${\displaystyle C\in \mathbb {R} ^{l\times n}}$

${\displaystyle \langle AB,C\rangle _{F}=\langle B,A^{T}C\rangle _{F}=\langle A,CB^{T}\rangle _{F}}$.

The same applies to the complex Frobenius scalar product for all and ${\displaystyle A\in \mathbb {C} ^{l\times m},B\in \mathbb {C} ^{m\times n}}$ ${\displaystyle C\in \mathbb {C} ^{l\times n}}$

${\displaystyle \langle AB,C\rangle _{F}=\langle B,A^{H}C\rangle _{F}=\langle A,CB^{H}\rangle _{F}}$.

Both properties result from the cyclical interchangeability of matrices under the track.

Invariances

Because of the trace display and the displacement property, the following applies to the real Frobenius scalar product of two matrices ${\displaystyle A,B\in \mathbb {R} ^{m\times n}}$

${\displaystyle \langle A,B\rangle _{F}=\langle A^{T},B^{T}\rangle _{F}}$.

The same applies to the complex Frobenius scalar product of two matrices ${\displaystyle A,B\in \mathbb {C} ^{m\times n}}$

${\displaystyle {\overline {\langle A,B\rangle _{F}}}=\langle A^{H},B^{H}\rangle _{F}}$.

Induced norm

The norm derived from the Frobenius scalar product is the Frobenius norm

${\displaystyle \|A\|_{F}=\left(\langle A,A\rangle _{F}\right)^{1/2}}$.

The Frobenius norm is thus particularly invariant under unitary transformations and the Cauchy-Schwarz inequality applies

${\displaystyle |\langle A,B\rangle _{F}|\leq \|A\|_{F}\,\|B\|_{F}}$.

The estimate then follows from this

${\displaystyle |\langle A,B\rangle _{F}|^{2}\leq \operatorname {spur} (A^{H}A)\cdot \operatorname {spur} (B^{H}B)}$,

where in the case of real matrices the adjoint is replaced by the transpose.

Estimation using the singular values

If the singular values of and those of with , then the estimate applies to the Frobenius scalar product ${\displaystyle \sigma _{1}(A),\ldots ,\sigma _{r}(A)}$ ${\displaystyle A}$ ${\displaystyle \sigma _{1}(B),\ldots ,\sigma _{r}(B)}$ ${\displaystyle B}$ ${\displaystyle r=\min\{m,n\}}$

${\displaystyle |\langle A,B\rangle _{F}|\leq \sum _{i=1}^{r}\sigma _{i}(A)\sigma _{i}(B)\leq \|A\|_{F}\,\|B\|_{F}}$,

This estimate represents an exacerbation of the Cauchy-Schwarz inequality above.

literature

• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 0-521-46713-6 .
• Roger A. Horn, Charles R. Johnson: Topics in Matrix Analysis . Cambridge University Press, 1994, ISBN 0-521-46713-6 .

Individual evidence

1. ^ Horn, Johnson: Matrix Analysis . S. 321 .
2. ^ Horn, Johnson: Topics in Matrix Analysis . S. 186 .

Web links

• pahio: Frobenius product . In: PlanetMath . (English)