Ela the Moore - Penrose Inverse of a Free Matrix

A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization relates the combinatorial structure of a free matrix to that of its Moore-Penrose inverse. Also, it is proved that the bipartite graph or, equivalently, the zero pattern of a free matrix uniquely determines that of its Moore-Penrose inverse, and this mapping is described explicitly. Finally, it is proved that a free matrix contains at most as many nonzero entries as does its Moore-Penrose inverse. 1. The main results. A set of complex numbers S is algebraically independent over the rational numbers Q if p(s 1 ,. .. , s n) = 0 whenever s 1 ,. .. , s n are distinct elements of S and p(x 1 ,. .. , x n) is a nonzero polynomial with rational coefficients. Lemma 1.1. Let S 1 and S 2 be finite sets of complex numbers so that each element of S 2 may be written as a rational form in elements of S 1 , and conversely. If S 1 is algebraically independent, then S 2 is algebraically independent if and only if |S 1 | = |S 2 |. Proof. For each finite set of complex numbers S, let dt Q S denote the maximal cardinality of an algebraically independent subset of S and note that dt Q S = dt Q Q(S). is algebraically independent, then |S 1 | = dt Q S 1 = dt Q Q(S 1) = dt Q Q(S 2) = dt Q S 2 , and the lemma follows. A matrix with complex entries is free, or generic, if the multiset of nonzero entries is algebraically independent. These nonzero entries may be viewed as indeterminants over the rational numbers; see [5, Chap. 6]. Free matrices have been used to represent objects from transversal theory, extremal poset theory, electrical network theory, and other combinatorial areas; see [3, Chap. 9] for a partial overview. The advantage of such representations is that they allow methods from linear algebra to be applied to combinatorial problems, most often via the connection given in Theorem 1.2 below. A nonzero partial diagonal of a matrix A is a collection of nonzero entries …