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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.

MSC:

15A04 Linear transformations, semilinear transformations
15A03 Vector spaces, linear dependence, rank, lineability
15A45 Miscellaneous inequalities involving matrices
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References:

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