Linear preservers for Sylvester and Frobenius bounds on matrix rank. (English) Zbl 1134.15003

Summary: Let \(A\) and \(B\) be \(n\times n\) 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
\[ \min\{\text{rank}(A), \text{rank}(B)\} \] and at least \(\text{rank}(A)+ \text{rank}(B)-n\). A generalization of Sylvester’s inequality is Frobenius’s inequality which states that
\[ \text{rank}(AB)+\text{rank}(BC) \leq \text{rank}(ABC)+\text{rank}(B). \]
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.


15A04 Linear transformations, semilinear transformations
15A03 Vector spaces, linear dependence, rank, lineability
15A45 Miscellaneous inequalities involving matrices
Full Text: DOI Euclid


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