A note on an algebraic version of Cochran's theorem (Q1881077)

Let \(M_n\) be the algebra of all complex \(n \times n\) matrices, \(n > 1\). The algebraic version of \textit{W. G. Cochran}'s theorem [Proc. Cambridge Philos. Soc. 30, 178--191 (1934; Zbl 0009.12004)] in statistics is linking the notions of orthogonal and rank additive family of matrices in \(M_n\): (i) A rank additive decomposition of an idempotent matrix is orthogonal. (ii) An orthogonal family of idempotent matrices is rank additive. The first part was extended before by \textit{P. Šemrl} [Linear Algebra Appl. 237--238, 477--487 (1996; Zbl 0847.62043)] for all matrices \(A\) having the property that every rank additive family with sum \(A\) is orthogonal. The author concludes this generalization extending the second part, i.e. characterizing all square matrices \(A\) for which every orthogonal family of matrices having \(A\) as its sum is rank additive.