On angles and distances between subspaces (Q734937)

The angles and distances between two given subspaces of \({\mathbb C}^{n,1}\) are investigated on the basis of a~joint decomposition of the corresponding orthogonal projectors. Several new results are established (in particular, conditions equivalent to \({\mathcal C}[\mathcal R(P),\mathcal R(Q)] = 0\) are given, corresponding characteristics of nilpotent, EP, and bi-EP matrices are obtained, alternative representation of the direct distance between \(\mathcal R(P)\) and~\(\mathcal R(Q)\) is found, a~relationship between the minimal angle and direct distance between \(P\) and~\(Q\) is derived, etc.), with the particular attention paid to the notions of inclinedness and minimal angle. Here, \(P, Q\in {\mathbb C}_n^{\mathrm{OP}} = \{N\in {\mathbb C}^{n,n}\,:\,N^2 = N =N^*\}\) are orthogonal projectors in~\({\mathbb C}^{n,1}\), \(\mathcal R(\cdot)\) is the column space, and \(\mathcal C(\mathcal M,\mathcal N)\) is the angle between subspace \(\mathcal M\) and~\(\mathcal N\). To demonstrate the usefulness of the approach utilized, some results known to be valid in Hilbert space are reestablished in \({\mathbb C}^{n,1}\), either in generalized form or with considerably shorter proofs than in the original sources.