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Maps preserving matrices of local reduced minimum modulus zero at a fixed vector.
- Source :
-
Linear Algebra & its Applications . Jan2024, Vol. 680, p65-82. 18p. - Publication Year :
- 2024
-
Abstract
- Let n be an integer greater than 1, and M n (C) be the algebra of all n × n -complex matrices. Let x 0 ∈ C n be a nonzero vector, and Φ be a linear map on M n (C) such that Φ (I) is invertible. For any matrix T ∈ M n (C) , let γ (T , x 0) denote the local reduced minimum modulus of T at x 0. In this paper, we show that Φ satisfies γ (T , x 0) = 0 ⇔ γ (Φ (T) , x 0) = 0 , (T ∈ M n (C)) , if and only if there are two invertible matrices A , B ∈ M n (C) such that A x 0 = A ⁎ x 0 = x 0 and Φ (T) = B T A for all T ∈ M n (C). When n = 2 , we show that the invertibility hypothesis of Φ (I) is redundant. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATRICES (Mathematics)
*LINEAR operators
*ALGEBRA
Subjects
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 680
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 173474451
- Full Text :
- https://doi.org/10.1016/j.laa.2023.09.022