Metric Frobenius norms and inner products of matrices and linear maps

The Frobenius norm is a frequent choice of norm for matrices. We provide a broader view on the Frobenius norm and Frobenius inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. These new concepts are termed the metric Frobenius norm and metric Frobenius inner product. We demonstrate that the classical Frobenius norm is merely one particular element of the family of metric Frobenius norms. We also show that the metric Frobenius norm has an interpretation similar to an operator norm of a linear map. While the usual operator norm is defined as the maximal norm response of the map w.r.t. inputs in the unit sphere, the Frobenius norm turns out to measure the average norm response.