The pyramid decomposition and rank minimization
The authors show that a general matrix \(M\) over a regular ring with unit admits a pyramid decomposition \(M=M_ 1 + M_ 2 + M_ 3 + M_ 4\) with \(M_ 1\leq M_ 1 + M_ 2 \leq M_ 1 + M_ 2 + M_ 3 \leq M\), where the so-called minus order \(A \leq B\) means that \(XA = XB\) and \(AY = BY\) for some inner inverses \(X\), \(Y\) of \(A\). In fact, there are many such decompositions derived from the Pierce decomposition of \(M\) relative to an arbitrary pair of idempotents.
scientific article; zbMATH DE number 459076