Linear operators that preserve pairs of matrices which satisfy extreme rank properties––a supplementary version
✍ Scribed by Xian Zhang
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 173 KB
- Volume
- 375
- Category
- Article
- ISSN
- 0024-3795
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✦ Synopsis
A pair of m × n matrices (A, B) is said to be rank-sum-maximal if ρ(A + B) = ρ(A) + ρ(B), and rank-sum-minimal if ρ(A + B) = |ρ(A)ρ(B)|. We characterize the linear operators preserving the set of rank-sum-maximal pairs over any field and the linear operators preserving the set of rank-sum-minimal pairs over any field except for {0, 1}. The linear preservers of the set of rank-sum-maximal pairs are characterized by using a result about rank preservers proposed by Li and Pierce [Amer. Math. Monthly 108 (2001) 591-605], and thereby the linear preservers of the set of rank-sum-minimal pairs are characterized. The paper can be viewed as a supplementary version of several related results.