## 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
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### References:

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