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 . 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

the Frobenius inner product is defined as

where the overline denotes the complex conjugate, and   denotes the Hermitian conjugate.^[1] Explicitly, this sum is

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, then the Frobenius inner product is the sum of the entries of the Hadamard product. If the matrices are vectorized (i.e., converted into column vectors, denoted by " "), then

Therefore

 ^{[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:

Also, exchanging the matrices amounts to complex conjugation:

For the same matrix,

 ,^{[citation needed]}

and,

 .

The inner product induces the Frobenius norm

 ^[1]

For two real-valued matrices, if

then

For two complex-valued matrices, if

then

while

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

• 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