Normalized Frobenius condition number of the orthogonal projections of the identity

This paper deals with the orthogonal projection (in the Frobenius sense) A N of the identity matrix I onto the matrix subspace A S ( A ∈ R n × n , S being an arbitrary subspace of R n × n ). Lower and upper bounds on the normalized Frobenius condition number of matrix A N are given. Furthermore, for every matrix subspace S ⊂ R n × n , a new index κ ̂ F ( A , S ) , which generalizes the normalized Frobenius condition number of matrix A , is defined and analyzed.