In mathematics, the Frobenius inner product (also known as the Double-dot product) is a binary operation that takes two matrices and returns a scalar. It is often denoted or . 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 dimensionâÂÂsame number of rows and columnsâÂÂbut are not restricted to be square matrices.
It is named after Ferdinand Georg Frobenius.
Given two complex-number-valued nÃÂm 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. Explicitly, this sum is
The calculation is very similar to the dot product of two vectors, which in turn is an example of an inner product.
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
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, the inner product induces the Frobenius norm
and is zero for a zero matrix,
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