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 formula_1. 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, and formula_4 denotes Hermitian 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 (without transpose) 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 (denoted by "vec", converted into column vectors) as follows,
the matrix product
reproduces the definition, therefore