2024/07/27 by Chi-Kwong Li, Ming-Cheng Tsai, Li, Chi-Kwong +5 · 3 citations
Computer Science · Mathematics · #15A60 #15A86 #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2408.06366
openalex publication_date 2024/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Two (real or complex) m× n matrices A and B are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm ‖⋅‖ if ‖A+ μB‖ = ‖A‖ + ‖B‖ for some scalar μ with |μ|=1 (resp. μ=1). We study linear maps T on m× n matrices preserving parallel (resp. TEA) pairs, i.e., T(A) and T(B) are parallel (resp. TEA) whenever A and B are parallel (resp. TEA). It is shown that when m,n ≥ 2 and (m,n) ≠ (2,2), a nonzero linear map T preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, T has the form (1) A ↦ γUAV or (2) A ↦ γUAt V (in this case, m = n), for a positive number γ, and unitary (or real orthogonal) matrices U and V of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form (3) A ↦ f(A) Z for a linear functional f and a fixed matrix Z. The case when (m,n) = (2,2) is more complicated. There are linear maps of 2× 2 matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups.