Chapter 4:algebraic sets, polynomials, and other functions

References

• Arazy , J. 1975 . The isometries of Cp . Israel J. Math. , 22 : 247 – 256 . (MR Nov. 79.)
   Google Scholar
• Atkinson , M. D. 1980 . Spaces of matrices with several zero eigenvalues . Bull. Lond. Math. Soc. , 12 : 89 – 95 .
   Google Scholar
• Beasley , L. 1970a . Linear transformations on matrices:The invariance of the third elementary symmetric function . Canad. J. Math. , 22 ( 4 ) : 746 – 752 . (MR Sept. 71.)
   Web of Science ®Google Scholar
• Beasley , L. 1978 . Linear transformations on matrices:The invariance of commuting pairs of matrices . Lin. And Mult. Alg. , 6 : 179 – 183 . (MR Jan. 80.)
   Google Scholar
• Botta , E. P. 1967a . Linear transformations that preserve the permanent . Proc. AMS , 18 : 566 – 569 . (MR Apr. 68.)
   Google Scholar
• Botta , E. P. 1967b . Linear transformations on matrices:The invariance of a class of general matrix function . Canad. J. Math. , 19 : 281 – 290 . (MR Feb. 68.)
   Google Scholar
• Botta , E. P. 1968a . Linear transformations on matrices:The invariance of a class of general matrix functions II . Canad. J. Math. , 20 : 739 – 748 . (MR Mar. 69.)
   Google Scholar
• Botta , E. P. 1968b . On the conversion of the determinant into the permanent . Canad. Math. Bull. , 11 : 31 – 34 . (MR June 69.)
   Web of Science ®Google Scholar
• Botta , E. P. 1978 . Linear maps that preserve singular and non-singular matrices . Lin. Alg. And Appl , 20 : 45 – 49 . (MR Aug. 79.)
   Web of Science ®Google Scholar
• Botta , E. P. 1979 . Linear transformations preserving the unitary group . Lin. and Mult. Alg. , 8 : 89 – 96 . (MR Aug. 81.)
   Google Scholar
• Botta , E. P. and Pierce , S. 1977 . The preservers of any orthogonal group . Pac. J. Math. , 104 : 347 – 359 . (MR JUne 79.)
   Google Scholar
• Botta , E. P. , Pierce , S. and Watkins , W. 1983 . Linear transformations that preserve the nilpotent matrices . Pac. J. Math , 104 : 39 – 46 . (MR Mar. 84.)
   Web of Science ®Google Scholar
• Chan , G. H. and Lim , M. H. 1982 . Linear transformations on symmetric matrices that preserve commutativity . Lin. Alg. and Appl. , 47 : 11 – 22 . (MR Nov. 83.)
   Web of Science ®Google Scholar
• Chan , G. H. , Lim , M. H. and Tan , K. K. 1987 . Linear preservers on matrices . Lin. Alg. and Appl. , 93 : 67 – 80 .
   Web of Science ®Google Scholar
• Choi , M. D. , Jafarian , A. A. and Radjavi , H. 1987 . Linear maps preserving commutativity . Lin. Alg. and Appl. , 87 : 227 – 241 . (MR Jun. 88.)
   Web of Science ®Google Scholar
• Dieudonne , J. 1949 . Sur une Generalisation du groupe orthogonal a quatre variables . Arch. Math. , 1 : 282 – 287 . (MR Oct. 49.)
   Google Scholar
• Dieudonne , J. 1951 . The automorphisms of the classical groups . AMS Memoirs , 2 (MR June 52.)
   Google Scholar
• Dixon , J. 1977 . Rigid embeddings of simple groups in the general linear group . Canad. J. Math. , 29 : 384 – 391 . (MR Apr. 78.)
   Web of Science ®Google Scholar
• Djokovic , D. Z. 1974 . Linear transformations on matrices . Trans. AMS , 198 : 93 – 106 . (MR May 76.)
   Google Scholar
• Dynkin , E. B. 1959 . The maximal subgroups of the classical groups . AMS Translation , 6 (MR Mar. 53.)
   Google Scholar
• Eaton , M. L. 1969 . On linear transformations which preserve the determinant . III. J. Math. , 13 : 722 – 727 . (Mr Oct. 70.)
   Google Scholar
• Frobenius , G. 1897 . Uber die Darstellung der endlichen Gruppen durch Lincare Substitutionen . S. B. Deutsch. Akad. Wiss. Berlin , 13 : 944 – 1015 .
   Google Scholar
• Gibson , P. M. 1974 . Unitary and orthogonal transformations on matrices . SIAM J. Appl. Math. , 27 ( 3 ) : 495 – 499 . (MR Oct. 75.)
   Web of Science ®Google Scholar
• Grone , R. 1980 . Certain isometrics of rectangular complex matrices . Lin. Alg. and Appl. , 29 : 161 – 171 . (MR Apr. 81.)
   Google Scholar
• Grone , R. and Marcus , M. 1977 . Isometries of matrix algebras . J. of Alg. , 47 : 180 – 189 . (MR Aug. 78.)
   Web of Science ®Google Scholar
• Howard , R. 1980 . Linear maps that preserve matrices annihilated by a polynomial . Lin. Alg. and Appl. , 30 : 167 – 176 . (MR Aug. 81.)
   Web of Science ®Google Scholar
• Hua , L. K. 1949 . Geometry of symmetric matrices over any field with characteristic other than two . Ann. of Math. , 2 ( 50 ) : 8 – 3l . (MR July 49.)
   Web of Science ®Google Scholar
• Hua , L. K. 1951 . A theorem on matrices over a field and its applications . J. Chinese Math. Soc. (N.S.) , 1 : 110 – 163 .
   Google Scholar
• James , D. G. 1981 . Linear transformations of the second elementary symmetric function . Lin. and Mult. Alg. , 10 : 347 – 349 . (MR Mar. 83.)
   Google Scholar
• Kapoor , J. 1972 . n -square matrices which, under row equivalence, leave the (n −)st elementary symmetric function invariant . Lin. Alg. and Appl. , 5 : 113 – 131 . (MR May 73.)
   Google Scholar
• Kovacs , A. 1976/1977 . Trace preserving linear transformations on matrix algebras . Lin. and Mult. Alg. , 4 : 243 – 250 . (MR Feb. 78.)
   Google Scholar
• Lauterman , C. 1981 . Linear transformations on matrices:Rank preservers and determinant preservers . Lin. and Mult. Alg. , 10 : 343 – 345 . (MR Jan. 83.)
   Google Scholar
• Marcus , M. 1959 . All linear operators leaving the unitary group invariant . Duke Math. J. , 26 : 155 – 163 . (MR Jan. 60.)
   Web of Science ®Google Scholar
• Marcus , M. and Gordon , W. 1970 . A generalization of the unitary group . Lin. Alg. and Appl. , 3 : 225 – 227 . (MR Aug. 71.)
   Google Scholar
• Marcus , M. and Holmes , J. 1972 . Groups of linear operators defined by group characters . Trans. AMS , 172 : 177 – 194 . (MR Dec. 73.)
   Google Scholar
• Marcus , M. and Khan , N. 1960 . A note on a group defined by a quadratic form . Can. Math. Bull. , 3 : 143 – 148 . (MR Mar. 62.)
   Google Scholar
• Marcus , M. and May , E. 1960 . On a theorem of I. Schur concerning matrix transformations . Archiv. D. Math. , 11 : 401 – 404 . (MR July 62.)
   Google Scholar
• Marcus , M. and Minc , H. 1961 . On the relation between the permanent and the determinant . III. J. Math. , 5 : 376 – 381 . (MR Nov. 63.)
   Google Scholar
• Marcus , M. and Minc , H. 1962 . The invariance of symmetric functions of singular values . Pac. J. Math. , 12 : 327 – 332 . (MR Nov. 63.)
   Google Scholar
• Marcus , M. , Minc , H. and Moyls , B. 1962 . Permanent preservers on the space of doubly-stochastic matrices . Can. J. Math. , 14 : 190 – 194 . (MR Jan. 63.)
   Google Scholar
• Marcus , M. and Moyls , B. 1959 . Transformations on tensor product spaces . Pac. J. Math. , 9 : 1215 – 1221 . (MR Dec. 60.)
   Google Scholar
• Marcus , M. and Newman , M. 1970 . Some results on unitary matrix groups . Lin. Alg. and Appl. , 3 : 173 – 178 . (MR May 71.)
   Google Scholar
• Marcus , M. and Purves , R. 1959 . Linear transformations on algebras of matrices II:The invariance of the elementary symmetric functions . Can. J. Math. , 11 : 383 – 396 . (MR July 60.)
   Google Scholar
• Marcus , M. and Westwick , R. 1960 . Linear maps on skew-symmetric matrices:The invariance of elementary . Pac. J. Math. , 10 : 917 – 924 . (MR July 61.)
   Google Scholar
• Minc , H. 1974 . Linear transformations on nonnegative matrices . Lin. Alg. and Appl. , 9 : 149 – 153 . (MR June 76.)
   Google Scholar
• Minc , H. 1976 . The invariance of the elementary symmetric functions . Lin. and Mult. Alg. , 4 : 209 – 215 . (MR Jan. 78.)
   Google Scholar
• Pierce , S. 1975 . Linear operators preserving the real symplectic group . Can. J. Math. , 27 : 715 – 724 . (MR Aug. 76.)
   Web of Science ®Google Scholar
• Pierce , S. 1979 . Discriminant preserving linear maps . Lin. and Mult. Alg. , 8 : 101 – 114 . (MR Nov. 80.)
   Google Scholar
• Pierce , S. and Watkins , W. 1978/1979 . Invariants of linear maps on matrix algebras . Lin. and Mult. Alg. , 6 : 185 – 200 . (MR Jun. 80.)
   Google Scholar
• Pierce , S. and Watkins , W. 1979 . Isometries of E2 . J. fur die Reine und Angew. Math. , 305 : 60 – 64 . (MR Aug. 80.)
   Google Scholar
• Prasolov , V. V. 1979 . Linear spaces in the general linear group . Geometric Methods in Problems of Algebra and Analysis , 2
   Google Scholar
• Rackusin , J. and Watkins , W. 1977 . Linear maps on matrices:The invariance of symmetric polynomials . Lin. Alg. and Appl. , 17 : 269 – 276 . (MR Jul. 79.)
   Web of Science ®Google Scholar
• Radjavi , H. 1984 . Commuiativiiy-preserving operators on symmetric matrices . Lin Alg. and Appl. , 61 : 219 – 224 . (MR Aug. 85.)
   Google Scholar
• Russo , B. 1969 . Trace preserving mappings of matrix algebras . Duke Math. J. , 36 : 297 – 300 . (MR Feb. 70.)
   Web of Science ®Google Scholar
• Schur , I. 1925 . Einige Bemerkungen zur Determinanten Theorie . Sitzungsberichtc der Preuss, Akad Wiss. Zu Berlin , 25 : 454 – 463 .
   Google Scholar
• Serezhkin , V. N. 1985 . Linear transformations preserving nilpotency . Vestsi Akad. Navuk BSSR. Ser. Fiz-Mat. Navuk , 6 ( 125 ) : 46 – 50 . (MR May 1987.)
   Google Scholar
• Waterhouse , W. C. 1983 . Invertibility of linear maps preserving matrix invariants . Lin. and Mult. Alg. , 13 : 105 – 113 . (MR Sep. 87.)
   Google Scholar
• Waterhouse , W. C. 1986 . Linear transformations on self-adjoint matrices:The preservation of rank-one-plus-scalar . Lin. Alg. and Appl. , 74 : 73 – 85 . (MR May 87.)
   Web of Science ®Google Scholar
• Watkins , W. 1976 . Linear maps that preserve commuting pairs of matrices . Lin. Alg. and Appl. , 14 : 29 – 35 . (MR Jul. 79.)
   Web of Science ®Google Scholar
• Watkins , W. 1982 . Linear transformations that preserve a similarity class of matrices . Lin. and Mult. Alg. , 11 : 19 – 22 . (MR Mar. 83.)
   Google Scholar
• Wei , A. 1975 . Linear transformations preserving the real orthogonal group . Can. J. Math. , 27 ( 3 ) : 561 – 572 . (MR Aug. 76.)
   Web of Science ®Google Scholar
• Wilde , A. C. 1986 . Transformations leaving the determinant of circulant matrices invariant . Amer. Math. Monthly , 93 : 356 – 361 . (MR Mar. 88.)
   Google Scholar
• Zassenhaus , H. J. 1958 . On a normal form of the orthogonal transformation . Can. Math. Bull. , 1 : 31 – 39 . (MR Feb. 60.)
   Google Scholar