In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar.
The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product.
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
If A and B are each real-valued matrices, then the Frobenius inner product is the sum of the entries of the Hadamard product.
"), 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, and, The inner product induces the Frobenius norm 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