The Moore idempotents of a matrix (Q1338507)

Let \({\mathbf R}\) be a commutative ring with 1 and an involution \(^ -\). Let \({\mathbf M}\) be the category of finite matrices over \({\mathbf R}\) with the involution \((a_{i,j}) \to (a_{i,j})^* = (\overline a_{j,i})\). If \(A \in {\mathbf M}\), then there exists a unique list of pairwise orthogonal symmetric idempotents \(e_ i\), \(i=0,\dots,s\), of \({\mathbf R}\) which are characterized by several properties involving rank and volume. In particular, \(A\) has a Moore-Penrose inverse in \({\mathbf M}\) iff \(e_ sA=0\).