## 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
Full Text:

### References:

 [1] L.B. Beasley, Linear operators which preserve pairs on which the rank is additive , J. Korean S.I.A.M. 2 (1998), 27-30. [2] L.B. Beasley and T.L. Laffey, Linear operators on matrices : The invariance of rank-$$k$$ matrices , Linear Algebra Appl. 133 (1990), 175-184. · Zbl 0721.15006 [3] L.B. Beasley, S.-G. Lee and S.-Z. Song, Linear operators that preserve pairs of matrices which satisfy extreme rank properties , Linear Algebra Appl. 350 (2002), 263-272. · Zbl 1004.15025 [4] P. Botta, Linear maps that preserve singular and nonsingular matrices , Linear Algebra Appl. 20 (1978), 45-49. · Zbl 0371.15005 [5] J. Dieudonné, Sur une generalisation du groupe orthogonal à quatre variables , Arch. Math. 1 (1949), 282-287. · Zbl 0032.10601 [6] Alexander Guterman, Linear preservers for matrix inequalities and partial orderings , Linear Algebra Appl. 331 (2001), 75-87. · Zbl 0985.15018 [7] M Marcus and R. Purves, Linear transformations on algebras of matrices II, The invariance of the elementary symmetric functions , Canad. J. Math. 11 (1959), 383-396. · Zbl 0086.01704
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.