Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

선형보존자 문제들에 관한 연구

  • 송석준 (제주대학교 자연과학대학 정보수학과)
  • Published : 2006.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

선형보존자 문제들은 행렬들로 구성되는 벡터공간들 사이에서 어떤 함수, 부분집합, 관계 등을 불변하게 옮기는 선형연산자의 형태를 규명하고 그와 동치가 되는 조건들을 찾는 연구주제들을 말한다. 이 논문에서는 선형보존자 문제에 대한 전반적인 연구문제들과 연구의 동기와 원인들, 활발한 연구주제들, 연구방법들 및 앞으로의 연구방향에 대하여 요약한다.

Keywords

References

  1. R. B. Bapat, S. Pati and S. Z. Song, Rank preservers of matrices over max algebra, Linear and Multilinear algebra 48 (2000), 149-164 https://doi.org/10.1080/03081080008818665
  2. L. Beasley, Linear transformations on matrices: The invariance of commuting pairs of matrices, Linear and Multilinear Algebra 6 (1978), 179-183 https://doi.org/10.1080/03081087808817236
  3. L. Beasley, Rank k-preservers and preservers of sets of ranks, Linear Algebra Appl. 55 (1983), 11-17 https://doi.org/10.1016/0024-3795(83)90163-5
  4. L. Beasley, Linear operators on matrices: The invariance of rank-k matrices, Linear Algebra Appl. 107 (1988), 161-167 https://doi.org/10.1016/0024-3795(88)90242-X
  5. L. Beasley, A. Guterman, S. G. Lee, and S. Z. Song, Linear transformations preserving the Grassmannian over Mn($Z_+$), Linear Algebra and Its Applications, 393 (2004), 39-46 https://doi.org/10.1016/j.laa.2003.08.018
  6. L. Beasley, A. Guterman, S. G. Lee, and S. Z. Song, Linear preservers of zeros of matrix polynomials, Linear Algebra and Its Applications 401 (2005), 325-340 https://doi.org/10.1016/j.laa.2004.08.015
  7. L. Beasley, A. Guterman, S. G. Lee, and S. Z. Song, Linear preservers of extremes of rank inequalities over semirings: Row and column ranks, Linear Algebra and Its Applications 413 (2006), 495-509 https://doi.org/10.1016/j.laa.2005.03.024
  8. L. Beasley, S. G. Lee, and S. Z. Song, Linear operators that preserve pairs of matrices which satisfy extreme rank properties, Linear Algebra and Its Applications 350 (2002), 263-271 https://doi.org/10.1016/S0024-3795(02)00293-8
  9. L. Beasley and N. Pullman, Nonnegative rank preserving operators, Linear Algebra Appl. 65 (1985), 207-223 https://doi.org/10.1016/0024-3795(85)90098-9
  10. L. Beasley and N. Pullman, Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77 https://doi.org/10.1016/0024-3795(84)90158-7
  11. L. Beasley and N. Pullman, Fuzzy rank-preserving operators, Linear Algebra Appl. 73 (1986), 197-211 https://doi.org/10.1016/0024-3795(86)90240-5
  12. L. Beasley and N. Pullman, Term-rank, permanent and rook polynomial preservers, Linear Algebra Appl. 90 (1987), 33-46 https://doi.org/10.1016/0024-3795(87)90302-8
  13. L. Beasley and N. Pullman, Semiring rank versus column rank, Linear Algebra Appl. 101 (1988), 33-48 https://doi.org/10.1016/0024-3795(88)90141-3
  14. L. Beasley and N. Pullman, Linear operators strongly preserving primitivity, Linear and Multilinear Algebra 25 (1989), 205-213 https://doi.org/10.1080/03081088908817942
  15. L. Beasley and N. Pullman, Linear operators preserving digraphs whose maximum cycle length is small, Linear and Multilinear Algebra 28 (1990), 111-117 https://doi.org/10.1080/03081089008818035
  16. L. Beasley and N. Pullman, Linear operators preserving properties of graphs, Congruss Numerantium 70 (1990), 105-112
  17. L. Beasley and N. Pullman, Linear operators strongly preserving commuting pairs of Boolean matrices, Linear Algebra Appl. 132 (1990), 137-143 https://doi.org/10.1016/0024-3795(90)90059-L
  18. L. Beasley and N. Pullman, Linear operators preserving idempotent matrices over fields, Linear Algebra Appl, 146 (1991), 7-20 https://doi.org/10.1016/0024-3795(91)90016-P
  19. L. Beasley and N. Pullman, Linear operators strongly preserving idempotent matrices over semirings, Linear Algebra Appl. 160 (1992), 217-229 https://doi.org/10.1016/0024-3795(92)90448-J
  20. L. Beasley and N. Pullman, Linear operators that strongly preserve graphical properties of matrices, Discrete Math. 104 (1992), 143-157 https://doi.org/10.1016/0012-365X(92)90329-E
  21. L. Beasley and S. G. Lee, Linear operators preserving r-potent matrices over semirings, Linear Algebra Appl. 164 (1992), 589-600 https://doi.org/10.1016/0024-3795(92)90394-P
  22. L. B. Beasley and S. Z. Song, A comparison of nonnegative real ranks and their preservers, Linear and Multilinear Algebra 31 (1992), 37-46 https://doi.org/10.1080/03081089208818120
  23. L. B. Beasley and S. Z. Song, Linear operators that preserve zero-term rank over fields and rings, Linear Algebra and Its Applications 341 (2002), 143-149 https://doi.org/10.1016/S0024-3795(01)00349-4
  24. L. Beasley, S. Z. Song, K. T. Kang, and B. Sarma, Column ranks and their preservers over nonnegative real matrices, Linear Algebra and Its Applications 399 (2005), 3-16 https://doi.org/10.1016/j.laa.2004.04.022
  25. L. B. Beasley, S. Z. Song and S. G. Lee, Linear operators that preserve zero-term rank of Boolean matrices, J. Korean Math. Soc. 36 (1999), no. 6, 1181-1190
  26. L. B. Beasley, S. Z. Song and S. G. Lee, Zero-term rank preservers, Linear and Multilinear Algebra 48 (2001), no. 4, 313-318 https://doi.org/10.1080/03081080108818677
  27. A. Berman, D. Hershkowitz, and C. R. Johnson, Linear transformations that preserve certain positivity classes of matrices, Linear Algebra Appl. 68 (1985), 9-29 https://doi.org/10.1016/0024-3795(85)90205-8
  28. E. P. Botta, Linear maps that preserve singular and nonsingular matrices, Linear Algebra Appl. 20 (1978), 45-49 https://doi.org/10.1016/0024-3795(78)90027-7
  29. E. P. Botta, Linear transformations preserving the unitary group, Linear and Multilinear Algebra 8 (1979), 89-96 https://doi.org/10.1080/03081087908817304
  30. E. P. Botta and S. Pierce, The preservers of any orthogonal group, Pacific J. Math. 70 (1977), 347-359 https://doi.org/10.2140/pjm.1977.70.347
  31. G. H. Chan and M. H. Lim, Linear transformations on symmetric matrices that preserve commutativity, Linear Algebra Appl. 47 (1982), 11-22 https://doi.org/10.1016/0024-3795(82)90222-1
  32. G. H. Chan and M. H. Lim, Linear transformations on tensor spaces, Linear and Multilinear Algebra 14 (1983), 3-9 https://doi.org/10.1080/03081088308817538
  33. G. H. Chan, M. H. Lim, and K. K. Tan, Linear preservers on powers of matrices, Linear Algebra Appl., 162 (1992), 615-626 https://doi.org/10.1016/0024-3795(92)90396-R
  34. G. H. Chan, M. H. Lim, and K. K. Tan, Linear preservers on matrices, Linear Algebra Appl. 93 (1987), 67-80 https://doi.org/10.1016/S0024-3795(87)90312-0
  35. M. D. Choi, A. A. Jafarian, and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227-241 https://doi.org/10.1016/0024-3795(87)90169-8
  36. J. Dieudonne, The Automorphisms of the Classical Groups, Mem. Amer. Math. Soc. 2 (1949)
  37. D. Z. Djokovic, Linear transformations of tensor products preserving a fixed rank, Pacific J. Math., 30 (1969), 411-414 https://doi.org/10.2140/pjm.1969.30.411
  38. G. Frobenius, Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. Berlin, (1897), pp. 994-1015
  39. R. Grone, Isometries of Matrix Algebras, Ph. D. Thesis, Univ. of California, Santa Barbara, (1976)
  40. R. Grone and M. Marcus, Isometries of matrix algebra, J. Algebra 47 (1977), 180-189 https://doi.org/10.1016/0021-8693(77)90218-6
  41. D. Hershkowitz and C. R. Johnson, Linear transformations which map the Pmatrices into themselves, Linear Algebra Appl. 74 (1986), 23-38 https://doi.org/10.1016/0024-3795(86)90113-8
  42. F. Hiai, Similarity preserving linear maps on matrices, Linear Algebra Appl. 97 (1987), 127-139 https://doi.org/10.1016/0024-3795(87)90145-5
  43. R. Horn, C. K. Li, and N. K. Tsing, Linear operators preserving certain equivalence relations on matrices, SIAM J. Matrix Anal. Appl. 12 (1991), 195-204 https://doi.org/10.1137/0612015
  44. S. G. Hwang, S. J. Kim and S. Z. Song, Linear operators that preserve maximal column rank of Boolean matrices, Linear Multilinear Algebra 36 (1994), no. 4, 305-313 https://doi.org/10.1080/03081089408818305
  45. C. R. Johnson and S. Pierce, Linear maps on hermitian matrices: The stabilizer of an inertia class II, Linear and Multilinear Algebra 19 (1986), 21-31 https://doi.org/10.1080/03081088608817701
  46. S. Kantor, Theorie der Aquivalenz von linearen $\infty$ Scharen bilinearer Formen, Sitzungsber. Munchener Akad., (1987), pp. 367-381
  47. S. Kirkland and N. J. Pullman, Linear operators preserving invariants of nonbinary matrices, Linear and Multilinear Algebra 33 (1992), 295-300 https://doi.org/10.1080/03081089308818200
  48. A. Kovacs, Trace preserving linear transformations on matrix algebras, Linear and Multilinear Algebra 4 (1976/77), 243-250 https://doi.org/10.1080/03081087708817158
  49. C. K. Li, Linear operators preserving the numerical radius of matrices, Proc. Amer. Math. Soc. 99 (1987), 601-608
  50. C. K. Li, L. Rodman, and N. K. Tsing, Linear operators preserving certain equivalence relations originating in system theory, Linear Algebra Appl. 161 (1992), 165-226 https://doi.org/10.1016/0024-3795(92)90011-X
  51. C. K. Li, B. S. Tam, and N. K. Tsing, Linear operators preserving the (p,q) numerical range, Linear Algebra Appl. 110 (1988), 75-89 https://doi.org/10.1016/0024-3795(83)90133-7
  52. C. K. Li and N. K. Tsing, 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 (1988), 353-362 https://doi.org/10.1080/03081088808817888
  53. C. K. Li and N. K. Tsing, Duality between some linear preserver problems.II. Isometries with re-spect to c-spectral norms and matrices with fixed singular values, Linear Algebra Appl. 110 (1988), 181-212
  54. C. K. Li and N. K. Tsing, Linear operators preserving unitarily invariant norms on matrices, Linear and Multilinear Algebra 26 (1990), 119-132 https://doi.org/10.1080/03081089008817969
  55. C. K. Li and N. K. Tsing, Linear operators preserving certain functions on singular values of ma-trices, Linear and Multilinear Algebra 26 (1990), 133-143 https://doi.org/10.1080/03081089008817970
  56. C. K. Li and N. K. Tsing, Linear operators preserving unitary similarity invariant norms on ma-trices, Linear and Multilinear Algebra 27 (1990), 213-224 https://doi.org/10.1080/03081089008818013
  57. C. K. Li and N. K. Tsing, Duality between some linear preserver problems. III. c-spectral norms on (skew)-symmetric matrices and matrixes with fixed singular values, Linear Algebra Appl. 143 (1991), 67-97
  58. C. K. Li and N. K. Tsing, Linear operators leaving a class of matrices with fixed singular values invariant, Linear and Multilinear Algebra, 34 (1993), 41-49 https://doi.org/10.1080/03081089308818207
  59. R. Loewy, Linear maps which preserve an inertia class, SIAM J. Matrix Anal. Appl. 11 (1990), 107-112 https://doi.org/10.1137/0611007
  60. R. Loewy and S. Pierce, Linear preservers of balanced singular inertia classes, Linear Algebra and its Applications, 201 (1994), 61-77 https://doi.org/10.1016/0024-3795(94)90105-8
  61. M. Marcus, Linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959), 155-163 https://doi.org/10.1215/S0012-7094-59-02615-8
  62. M. Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837-847 https://doi.org/10.2307/2311230
  63. M. Marcus, Linear transformations on matrices, J. Res. Nat. Bur. Standards 75B (1971), 107-113
  64. M. Marcus and H. Mine, On the relation between the permanent and the determinant, Illinois J. Math. 5 (1962), 327-332
  65. M. Marcus and B. Moyls, Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215-1221 https://doi.org/10.2140/pjm.1959.9.1215
  66. S. Pierce, Linear maps on algebraic groups, Linear Algebra Appl. 162 (1992), 237-242 https://doi.org/10.1016/0024-3795(92)90378-N
  67. S. Pierce and W. Watkins, Invariants of linear maps on matrix algebras, Linear and Multilinear Algebra 6 (1989/79), 185-200
  68. S. Pierce et al., A survey of linear preserver problems, Linear and Multilinear Algebra 33 (1992), 1-119 https://doi.org/10.1080/03081089208818176
  69. G. Polya, Aufgabe 424, Arch. Math. u. phys. 203 (1913), 271
  70. H. Radjavi, Commutativity-preserving operators on symmetric matrices, Linear Algebra Appl. 61 (1984), 219-224 https://doi.org/10.1016/0024-3795(84)90032-6
  71. R. Sinkhorn, Linear adjugate preservers on complex matrices, Linear and Multilinear Algebra 12 (1982/83), 215-222 https://doi.org/10.1080/03081088208817485
  72. S. Z. Song, Linear operators that preserve column rank of Boolean matrices, Proc. Amer. Math. Soc. 119(4) (1993), 1085-1088
  73. S. Z. Song, Linear operators that preserve column rank of fuzzy matrices, Fuzzy Sets and Systems 62 (1994), 311-317 https://doi.org/10.1016/0165-0114(94)90115-5
  74. S. Z. Song, On spanning column rank of matrices over semirings, Bull. Korean Math. Soc. 32 (1995), 337-342
  75. S. Z. Song, A comparison of maximal column ranks of matrices over related semirings, J. Korean Math. Soc. 34 (1997), no. 1, 213-225
  76. S. Z. Song, Linear operators that preserve maximal column ranks of nonnegative integer matrices, Proc. Amer. Math. Soc. 126 (1998), 2205-2211
  77. S. Z. Song, L. Beasley, G. S. Cheon and Y. B. Jun, Rank and perimeter preserver of Boolean rank-1 matrices, J. Korean Math. Soc. 41 (2004), no. 2, 397-406 https://doi.org/10.4134/JKMS.2004.41.2.397
  78. S. Z. Song and S. G. Hwang, Spanning column ranks and their preservers of nonnegative matrices, Linear Algebra and Its Applications 254 (1997), 485-495 https://doi.org/10.1016/S0024-3795(96)00514-9
  79. S. Z. Song and K. T. Kang, Column ranks and their preservers of matrices over max algebra, Linear and Multilinear Algebra 51 (2003), no. 3, 311-318 https://doi.org/10.1080/0308108031000069173
  80. S. Z. Song and K. T. Kang, Linear maps that preserve commuting pairs of matrices over general Boolean algebra, J. Korean Math. Soc. 43 (2006), no. 1, 77-86 https://doi.org/10.4134/JKMS.2006.43.1.077
  81. S. Z. Song, K. T. Kang and Y. B. Jun, Linear preservers of Boolean nilpotent matrices, J. Korean Math. Soc. 43 (2006), no. 3, 539-552 https://doi.org/10.4134/JKMS.2006.43.3.539
  82. S. Song and S. Lee, Column ranks and their preservers of general Boolean matrices, J. Korean Math. Soc. 32 (1995), 531-540
  83. S. Z. Song and S. R. Park, Maximal column rank preservers of fuzzy matrices, Discussiones Mathematicae - General Algebra and Applications 21 (2001), no. 2, 207-218 https://doi.org/10.7151/dmgaa.1038
  84. S. Z. Song, S. D. Yang, S. M. Hong, Y. B. Jun and S. J. Kim, Linear operators preserving maximal column ranks of nonbinary Boolean matrices, Discussiones Mathematicae-General Algebra and Applications 20 (2000), no. 2, 255-265 https://doi.org/10.7151/dmgaa.1021
  85. W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14 (1976), 29-35 https://doi.org/10.1016/0024-3795(76)90060-4

Cited by

  1. Extreme Preservers of Zero-term Rank Sum over Fuzzy Matrices vol.50, pp.4, 2010, https://doi.org/10.5666/KMJ.2010.50.4.465
  2. EXTREME PRESERVERS OF FUZZY MATRIX PAIRS DERIVED FROM ZERO-TERM RANK INEQUALITIES vol.33, pp.3, 2011, https://doi.org/10.5831/HMJ.2011.33.3.301