Linear Preservers for Sylvester and Frobenius Bounds on Matrix Rank
LeRoy B. Beasley, Alexander Emilevich Guterman, Cora L. Neal · Rocky Mountain Journal of Mathematics · 2006
Let A and B be n × n matrices.A classical result about the rank function is Sylvester's inequality which states that the rank of the product of AB is at most min{rank (A), rank (B)} and at least rank (A) + rank (B) -n.A generalization of Sylvester's inequality is Frobenius's inequality which states that rank (AB) + rank (BC) ≤ rank (ABC) + rank (B).In this paper we investigate the structure of linear operators that preserve those ordered pairs or triples of matrices which satisfy one of the extreme cases in these inequalities.2000