Frobenius inner product


In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a number. It is often denoted. The operation is a component-wise inner product of two matrices as though they are vectors. The two matrices must have the same dimension—same number of rows and columns—but are not restricted to be square matrices.

Definition

Given two complex number-valued n×m matrices A and B, written explicitly as
the Frobenius inner product is defined by the following summation Σ of matrix elements,
where the overline denotes the complex conjugate. Explicitly this sum is
The calculation is very similar to the dot product, which in turn is an example of an inner product.

Properties

It is a sesquilinear form, for four complex-valued matrices A, B, C, D, and two complex numbers a and b:
Also, exchanging the matrices amounts to complex conjugation:
For the same matrix,

Examples

Real-valued matrices

For two real-valued matrices, if
then

Complex-valued matrices

For two complex-valued matrices, if
then the complex conjugates are
and
while
The Frobenius inner products of A with itself, and B with itself, are respectively

Frobenius norm

The inner product induces the Frobenius norm

Relation to other products

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 as follows,
the matrix product
reproduces the definition, therefore