Normalized Frobenius condition number of the orthogonal projections of the identity

This paper deals with the orthogonal projection (in the Frobenius sense) ANAN of the identity matrix II onto the matrix subspace ASAS (A∈Rn×nA∈Rn×n, SS being an arbitrary subspace of Rn×nRn×n). Lower and upper bounds on the normalized Frobenius condition number of matrix ANAN are given. Furthermore, for every matrix subspace S⊂Rn×nS⊂Rn×n, a new index κ̂F(A,S), which generalizes the normalized Frobenius condition number of matrix AA, is defined and analyzed.