Normalized Frobenius condition number of the orthogonal projections of the identity (Q1941107)

The authors deal with the orthogonal projection \(AN\) (in the Frobenius sense) of the identity matrix \(I\) onto the matrix subspace \(AS\) (\(A\in \mathbb R^{n\times n}\), \(S\) being an arbitrary subspace of \(\mathbb R^{n\times n}\)). Lower and upper bounds on the normalized Frobenius condition number of the matrix \(AN\) are given. Further, for every matrix subspace \(S\subset \mathbb R^{n\times n}\), a new index \(\hat k_F(A,S)\), which generalizes the normalized Frobenius condition number of the matrix \(A\), is defined and analyzed.