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 as, 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. 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 (i.e., converted into column vectors, denoted by "formula_6"), then Therefore 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, Frobenius norm. The inner  product induces the Frobenius norm Examples. Real-valued matrices. For two real-valued matrices, if then Complex-valued matrices. For two complex-valued matrices, if then while The Frobenius inner products of A with itself, and B with itself, are respectively