Rational solutions of certain matrix equations

• Published in: Linear algebra and its applications Vol. 430; no. 2; pp. 660 - 663
• Main Authors: Arav, Marina, Hall, Frank, Li, Zhongshan, Rao, Bhaskara
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.01.2009; Elsevier
• Subjects: Algebra; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Matrix equations; Minimum rank; Rational solution; Sciences and techniques of general use; Sign pattern; Minimum rank; 15A24; 11D09; Sign pattern; 11P21; Matrix equations; 15A48; 15A36; 15A03; Rational solution; Forests; Operator equation; Integer matrix; Positive definite matrix; Sign; Bipartite graph; Matrix equation; Rational function; Real matrix
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.09.011

A sign pattern is a matrix whose entries are from the set { + , - , 0 } . For a real matrix B, sgn ( B ) is the sign pattern obtained by replacing each positive (respectively, negative, zero) entry of B by + (respectively, - , 0). For a sign pattern A, the sign pattern class of A, denoted Q ( A ) , is defined as { B : sgn ( B ) = A } . The purpose of this note is to prove the following theorem: Let A, B, and C be real matrices such that AB = C . Suppose that the zero bipartite graph of C (this is the same as the complement of the bipartite graph of C) is a forest. Then there exist rational perturbations A ∼ , B ∼ and C ∼ of A , B , and C, respectively, in the same corresponding sign pattern classes, such that A ∼ B ∼ = C ∼ .