×

Linear preserver problems: A brief introduction and some special techniques. (English) Zbl 0762.15016

Let \(M\) be any one of the following matrix spaces: the set of all \(m\times n\) matrices over the field \(\mathbb{F}\), where usually \(\mathbb{F}\) is \(\mathbb{R}\) or \(\mathbb{C}\); the set of all \(n\times n\) symmetric matrices over \(\mathbb{F}\); the set of all \(n\times n\) skew-symmetric matrices over \(\mathbb{F}\); the set of all Hermitian matrices. The typical linear preserving problems are:
1. let \(F\) be a (scalar-valued, vector-valued, or set-valued) function on \(M\). Characterize those linear operators \(\varphi\) on \(M\) that satisfy \(F(\varphi(A))=F(A)\) for all \(A\in M\);
2. let \(S\subset M\). Characterize those linear operators \(\varphi\) on \(M\) that satisfy \(\varphi(S)=S\) or \(\subset S\);
3. let \(\sim\) be a relation or an equivalence relation on \(M\). Characterize those linear operators \(\varphi\) on \(M\) that satisfy \(\varphi(A)\sim\varphi(B)\) whenever \(A\sim B\) (or iff \(A\sim B)\);
4. given a transform \(F:M\to M\), characterize those linear operators \(\varphi\) on \(M\) that satisfy \(F(\varphi(A))=\varphi(F(A))\) for all \(A\in M\).
This paper is a survey which gives a gentle introduction to these problems.
Reviewer: V.L.Popov (Moskva)

MSC:

15A72 Vector and tensor algebra, theory of invariants
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15B57 Hermitian, skew-Hermitian, and related matrices
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Beasley, L., Linear transformations on matrices: The invariance of commuting pairs of matrices, Linear and Multilinear Algebra, 6, 179-183 (1978) · Zbl 0397.15010
[2] Beasley, L., Rank \(k\)-preservers and preservers of sets of ranks, Linear Algebra Appl., 55, 11-17 (1983) · Zbl 0526.15004
[3] Beasley, L., Linear operators on matrices: The invariance of rank-\(k\) matrices, Linear Algebra Appl., 107, 161-167 (1988) · Zbl 0651.15004
[4] Beasley, L.; Pullman, N., Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl., 59, 55-77 (1984) · Zbl 0536.20044
[5] Beasley, L.; Pullman, N., Fuzzy rank-preserving operators, Linear Algebra Appl., 73, 197-211 (1986) · Zbl 0578.15002
[6] Berman, A.; Hershkowitz, D.; Johnson, C. R., Linear transformations that preserve certain positivity classes of matrices, Linear Algebra Appl., 68, 9-29 (1985) · Zbl 0583.15013
[7] Botta, E. P., Linear maps that preserve singular and nonsingular matrices, Linear Algebra Appl., 20, 45-49 (1978) · Zbl 0371.15005
[8] Botta, E. P., Linear transformations preserving the unitary group, Linear and Multilinear Algebra, 8, 89-96 (1979) · Zbl 0435.15004
[9] Botta, E. P.; Pierce, S., The preservers of any orthogonal group, Pacific J. Math., 70, 347-359 (1977) · Zbl 0381.15004
[10] Chan, G. H.; Lim, M. H., Linear transformations on symmetric matrices that preserve commutativity, Linear Algebra Appl., 47, 11-22 (1982) · Zbl 0492.15006
[11] Chan, G. H.; Lim, M. H., Linear transformations on tensor spaces, Linear and Multilinear Algebra, 14, 3-9 (1983) · Zbl 0522.15015
[13] Chan, G. H.; Lim, M. H.; Tan, K. K., Linear preservers on matrices, Linear Algebra Appl., 93, 67-80 (1987) · Zbl 0619.15003
[14] Choi, M. D.; Jafarian, A. A.; Radjavi, H., Linear maps preserving commutativity, Linear Algebra Appl., 87, 227-241 (1987) · Zbl 0615.15004
[15] Dieudonné, J., The Automorphisms of the Classical Groups, Mem. Amer. Math. Soc., 2 (1949)
[16] Djokovic, D. Z., Linear transformations of tensor products preserving a fixed rank, Pacific J. Math., 30, 411-414 (1969) · Zbl 0185.08302
[17] Frobenius, G., Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. Berlin, 994-1015 (1897) · JFM 28.0130.01
[18] Grone, R., Isometries of Matrix Algebras, (Ph.D. Thesis (1976), Univ. of California: Univ. of California Santa Barbara) · Zbl 0359.15012
[19] Grone, R.; Marcus, M., Isometries of matrix algebra, J. Algebra, 47, 180-189 (1977) · Zbl 0359.15012
[20] Hershkowitz, D.; Johnson, C. R., Linear transformations which map the \(P\)-matrices into themselves, Linear Algebra Appl., 74, 23-38 (1986) · Zbl 0591.15001
[21] Hiai, F., Similarity preserving linear maps on matrices, Linear Algebra Appl., 97, 127-139 (1987) · Zbl 0635.15002
[23] Johnson, C. R.; Pierce, S., Linear maps on hermitian matrices: The stabilizer of an inertia class II, Linear and Multilinear Algebra, 19, 21-31 (1986) · Zbl 0593.15021
[24] Kantor, S., Theorie der Äquivalenz von linearen ∞ Scharen bilinearer Formen, Sitzungsber. Münchener Akad., 367-381 (1987)
[25] Kovacs, A., Trace preserving linear transformations on matrix algebras, Linear and Multilinear Algebra, 4, 243-250 (1976/77) · Zbl 0361.15012
[26] Li, C. K., Linear operators preserving the numerical radius of matrices, Proc. Amer. Math. Soc., 99, 601-608 (1987) · Zbl 0627.15010
[28] Li, C. K.; Tam, B. S.; Tsing, N. K., Linear operators preserving the (p,q) numerical range, Linear Algebra Appl., 110, 75-89 (1988) · Zbl 0655.15025
[29] Li, C. K.; Tsing, N. K., Duality between some linear preserver problems: The invariance of the \(c\)-numerical range, the \(c\)-numerical radius and certain matrix sets, Linear and Multilinear Algebra, 23, 353-362 (1988) · Zbl 0668.15014
[30] Li, C. K.; Tsing, N. K., Duality between some linear preserver problems II. Isometries with respect to \(c\)-spectral norms and matrices with fixed singular values, Linear Algebra Appl., 110, 181-212 (1988) · Zbl 0655.15026
[31] Li, C. K.; Tsing, N. K., Linear operators preserving unitarily invariant norms on matrices, Linear and Multilinear Algebra, 26, 119-132 (1990) · Zbl 0691.15006
[32] Li, C. K.; Tsing, N. K., Linear operators preserving certain functions on singular values of matrices, Linear and Multilinear Algebra, 26, 133-143 (1990) · Zbl 0703.15024
[33] Li, C. K.; Tsing, N. K., Linear operators preserving unitary similarity invariant norms on matrices, Linear and Multilinear Algebra, 27, 213-224 (1990) · Zbl 0706.15028
[34] Li, C. K.; Tsing, N. K., Duality between some linear preserver problems. III. \(c\)-spectral norms on (skew)-symmetric matrices and matrices with fixed singular values, Linear Algebra Appl., 143, 67-97 (1991) · Zbl 0712.15028
[36] Loewy, R., Linear maps which preserve an inertia class, SIAM J. Matrix Anal. Appl., 11, 107-112 (1990) · Zbl 0697.15003
[38] Marcus, M., All linear operators leaving the unitary group invariant, Duke Math. J., 26, 155-163 (1959) · Zbl 0084.01701
[39] Marcus, M., Linear operations on matrices, Amer. Math. Monthly, 69, 837-847 (1962) · Zbl 0108.01104
[40] Marcus, M., Linear transformations on matrices, J. Res. Nat. Bur. Standards, 75B, 107-113 (1971) · Zbl 0244.15013
[41] Marcus, M.; Minc, H., On the relation between the permanent and the determinant, Illinois J. Math., 5, 327-332 (1962)
[42] Marcus, M.; Moyls, B., Transformations on tensor product spaces, Pacific J. Math., 9, 1215-1221 (1959) · Zbl 0089.08902
[44] Pierce, S.; Watkins, W., Invariants of linear maps on matrix algebras, Linear and Multilinear Algebra, 6, 185-200 (1978/79) · Zbl 0397.15011
[45] Polya, G., Aufgabe 424, Arch. Math. u. Phys., 203, 271 (1913)
[46] Radjavi, H., Commutativity-preserving operators on symmetric matrices, Linear Algebra Appl., 61, 219-224 (1984) · Zbl 0547.15007
[47] Sinkhorn, R., Linear adjugate preservers on complex matrices, Linear and Multilinear Algebra, 12, 215-222 (1982/83) · Zbl 0496.15003
[49] Wong, W. J., Maps on spaces of linear transformations, Math. Chronicle, 16, 15-24 (1987) · Zbl 0644.20011
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.