Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-07T12:08:36.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Mappings on matrices: invariance of functional values of matrix products

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

Jor-Ting Chan ,
Chi-Kwong Li  and
Nung-Sing Sze
Jor-Ting Chan
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, e-mail: jtchan@hku.hk
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA, e-mail: ckli@math.wm.edu
Nung-Sing Sze
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, and Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA, e-mail: sze@math.uconn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.

Keywords

zero product preserversunitary similarity invariant functions.

MSC classification

Secondary: 15A04: Linear transformations, semilinear transformations 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18: Eigenvalues, singular values, and eigenvectors
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 165 - 184
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alpers, B. and Schröder, E. M., ‘On mappings preserving orthogonality of nonsingular vectors’, J. Geom. 41 (1991), 3–15.CrossRefGoogle Scholar
[2]Bhatia, R., Šemrl, P. and Sourour, A., ‘Maps on matrices that preserve the spectral radius distance’, Studia Math. 134 (1999), 99–110.Google Scholar
[3]Chebotar, M. A., Ke, W.-F., Lee, P.-H. and Wong, N.-C., ‘Mappings preserving zero products’, Studia Math. 155 (2003), 77–94.CrossRefGoogle Scholar
[4]Cui, J. and Hou, J., ‘Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras’, J. Funct. Anal. 206 (2004), 414–448.CrossRefGoogle Scholar
[5]Dolinar, G. and Šemrl, P., ‘Determinant preserving maps on matrix algebras’, Linear Algebra Appl. 348 (2002), 189–192.Google Scholar
[6]Hou, J., ‘Maps preserving numerical range of operator products and Jordan triple products’, presentation at the mini-workshop on preserver problems, University of Hong Kong, 07, 2004.Google Scholar
[7]Hou, J. and Zhang, X., ‘Ring isomorphisms and linear or additive maps preserving zero products on nest algebras’, Linear Algebra Appl. 387 (2004), 343–360.Google Scholar
[8]Hua, L.-K., ‘A theorem on matrices over a sfield and its applications’, J. Chinese Math. Soc. (N.S.) 1 (1951), 109–163.Google Scholar
[9]Kestelman, H., ‘Automorphisms of the field of complex numbers’, Proc. London Math. Soc. (2) 53 (1951), 1–12.Google Scholar
[10]Li, C-K., ‘C-numerical ranges and C-numerical radii’, Linear and MultilinearAlgebra 37 (1994), 51–82.CrossRefGoogle Scholar
[11]Li, C.-K. and Pierce, S., ‘Linear preserver problems’, Amer. Math. Monthly 108 (2001), 591–605.Google Scholar
[12]Molnár, L., ‘Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn's version of Wigner's theorem’, J. Funct. Anal. 194 (2002), 248–262.CrossRefGoogle Scholar
[13]Molnár, L., ‘Some characterizations of the automorphisms of B(H) and C(X)’, Proc. Amer. Math. Soc. 130 (2002), 111–120.CrossRefGoogle Scholar
[14]Molnár, L., ‘Local automorphisms of operator algebras on Banach spaces’, Proc. Amer. Math. Soc. 131 (2003), 1867–1874.CrossRefGoogle Scholar
[15]Pierce, S., editor of the special issue on ‘A survey of linear preserver problems’, Linear and Multilinear Algebra 33 (1992).Google Scholar
[16]Tan, V. and Wang, F., ‘On determinant preserver problems’, Linear Algebra Appl. 369 (2003), 311–317.CrossRefGoogle Scholar
[17]Uhlhorn, U., ‘Representation of symmetry transformations in quantum mechanics’, Ark. Fys. 23 (1963), 307–340.Google Scholar
[18]Šemrl, P., ‘Linear mappings preserving square-zero matrices’, Bull. Austral. Math. Soc. 48 (1993), 365–370.Google Scholar
[19]Šemrl, P., ‘Applying projective geometry to transformations on rank-one idempotents’, J. Funct Anal. 210 (2004), 248–257.Google Scholar
[20]Šemrl, P., ‘Maps on matrix spaces’, Taussky-Todd Lecture, International Linear Algebra Society Conference, Coimbra, Portugal, 2004.Google Scholar
[21]Šemrl, P., ‘Maps on idempotents’, preprint.Google Scholar
[22]Yale, P. B., ‘Automorphisms of the complex numbers’, Math. Mag. 39 (1966), 135–141.Google Scholar