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.

DefinitionEdit

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 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.

Relation to other productsEdit

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 " "), then

  

Therefore

 

PropertiesEdit

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,

 

Frobenius normEdit

The inner product induces the Frobenius norm

 

ExamplesEdit

Real-valued matricesEdit

For two real-valued matrices, if

 

then

 

Complex-valued matricesEdit

For two complex-valued matrices, if

 

then

 

while

 

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

  

See alsoEdit