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Volumn 406, Issue 1-3, 2005, Pages 1-76

Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence

Author keywords

Canonical forms; Congruence; Indefinite inner product; Matrix pairs; Skew symmetric matrices; Strict equivalence

Indexed keywords

COMPUTER SIMULATION; HAMILTONIANS; LARGE SCALE SYSTEMS; MATHEMATICAL MODELS; NUMERICAL METHODS; QUADRATIC PROGRAMMING;

EID: 23144446628     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2005.03.035     Document Type: Article
Times cited : (56)

References (62)
  • 1
    • 0042252516 scopus 로고
    • Realization and factorization of rational matrix functions with symmetries
    • D. Alpay, J.A. Ball, I. Gohberg, and L. Rodman Realization and factorization of rational matrix functions with symmetries Oper. Theor.: Adv. Appl. 47 1990 1 60
    • (1990) Oper. Theor.: Adv. Appl. , vol.47 , pp. 1-60
    • Alpay, D.1    Ball, J.A.2    Gohberg, I.3    Rodman, L.4
  • 2
    • 0037289010 scopus 로고    scopus 로고
    • Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils
    • P. Benner, R. Byers, V. Mehrmann, and H. Xu Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils SIAM J. Matrix Anal. Appl. 24 2002 165 190
    • (2002) SIAM J. Matrix Anal. Appl. , vol.24 , pp. 165-190
    • Benner, P.1    Byers, R.2    Mehrmann, V.3    Xu, H.4
  • 3
    • 0000973369 scopus 로고    scopus 로고
    • A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems
    • P. Benner, V. Mehrmann, and H. Xu A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems Electron. Trans. Numer. Anal. 8 1999 115 126
    • (1999) Electron. Trans. Numer. Anal. , vol.8 , pp. 115-126
    • Benner, P.1    Mehrmann, V.2    Xu, H.3
  • 4
    • 84956092860 scopus 로고
    • The normal form of a Hamiltonian system
    • A.D. Bryuno The normal form of a Hamiltonian system Uspekhi Mat. Nauk 43 1988 23 56 (in Russian). English translation: Russian Math. Surveys 43 (1988) 25-66.
    • (1988) Uspekhi Mat. Nauk , vol.43 , pp. 23-56
    • Bryuno, A.D.1
  • 5
    • 0001252279 scopus 로고
    • Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues
    • N. Burgoyne, and R. Cushman Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues Celest. Mech. 8 1973/1974 435 443
    • (1973) Celest. Mech. , vol.8 , pp. 435-443
    • Burgoyne, N.1    Cushman, R.2
  • 6
    • 0002543105 scopus 로고
    • Normal forms for real linear Hamiltonian systems
    • Math. Sci. Press, Brookline, MA
    • Normal forms for real linear Hamiltonian systems, Lie Groups: History Frontiers and Applications VII (1977) 483-529, Math. Sci. Press, Brookline, MA.
    • (1977) Lie Groups: History Frontiers and Applications , vol.7 , pp. 483-529
  • 7
    • 0032615706 scopus 로고    scopus 로고
    • A unified approach to linear and nonlinear normal forms for Hamiltonian systems
    • R.C. Churchill, and M. Kummer A unified approach to linear and nonlinear normal forms for Hamiltonian systems J. Symbolic Comput. 27 1998 49 131
    • (1998) J. Symbolic Comput. , vol.27 , pp. 49-131
    • Churchill, R.C.1    Kummer, M.2
  • 8
    • 23144432116 scopus 로고    scopus 로고
    • Some properties of elementary Hamiltonian matrices
    • R. Coleman Some properties of elementary Hamiltonian matrices J. Math. Sci. (New York) 108 2002 951 965 (Translation from Russian original 1999).
    • (2002) J. Math. Sci. (New York) , vol.108 , pp. 951-965
    • Coleman, R.1
  • 9
    • 23144447477 scopus 로고
    • On the solutions of quadratic matrix equations
    • A.N. Čurilov On the solutions of quadratic matrix equations Non-Linear Vibrat. Control Theor. 2 1978 24 33 (in Russian).
    • (1978) Non-Linear Vibrat. Control Theor. , vol.2 , pp. 24-33
    • Čurilov, A.N.1
  • 10
    • 0000865355 scopus 로고
    • Sur la reduction canonique des couples de matrices
    • J. Dieudonné Sur la reduction canonique des couples de matrices Bull. Soc. Math. France 74 1946 130 146
    • (1946) Bull. Soc. Math. France , vol.74 , pp. 130-146
    • Dieudonné, J.1
  • 11
    • 36749118959 scopus 로고
    • Normal forms of elements of classical real and complex Lie and Jordan algebras
    • D.Ž. Djoković, J. Patera, P. Winternitz, and H. Zassenhaus Normal forms of elements of classical real and complex Lie and Jordan algebras J. Math. Phys. 24 1983 1363 1374
    • (1983) J. Math. Phys. , vol.24 , pp. 1363-1374
    • Djoković, D.Ž.1    Patera, J.2    Winternitz, P.3    Zassenhaus, H.4
  • 12
    • 23144437044 scopus 로고
    • Simultaneous reduction of a pair of bilinear forms to canonical form
    • J.B. Ermolaev Simultaneous reduction of a pair of bilinear forms to canonical form Dokl. Akad. Nauk SSSR 132 1960 257 259 (in Russian). Translation: Soviet Math. Dokl. 1 (1960) 523-525.
    • (1960) Dokl. Akad. Nauk SSSR , vol.132 , pp. 257-259
    • Ermolaev, J.B.1
  • 13
    • 0022681247 scopus 로고
    • Algebraic Riccati equations and symplectic algebra
    • L.E. Faibusovich Algebraic Riccati equations and symplectic algebra Int. J. Control 43 1986 781 792
    • (1986) Int. J. Control , vol.43 , pp. 781-792
    • Faibusovich, L.E.1
  • 15
    • 19944397710 scopus 로고
    • On Hamiltonian rational transfer functions
    • P.A. Fuhrmann On Hamiltonian rational transfer functions Linear Algebra Appl. 63 1984 1 93
    • (1984) Linear Algebra Appl. , vol.63 , pp. 1-93
    • Fuhrmann, P.A.1
  • 20
    • 23144444630 scopus 로고
    • On the theory of automorphic functions of a matrix variable, II-The classification of hypercircles under the symplectic group
    • L.K. Hua On the theory of automorphic functions of a matrix variable, II-The classification of hypercircles under the symplectic group Amer. J. Math 66 1944 531 563
    • (1944) Amer. J. Math , vol.66 , pp. 531-563
    • Hua, L.K.1
  • 21
    • 23144465852 scopus 로고
    • The equivalence of pairs of Hermitian matrices
    • M.H. Ingraham, and K.W. Wegner The equivalence of pairs of Hermitian matrices Trans. Amer. Math. Soc. 38 1935 145 162
    • (1935) Trans. Amer. Math. Soc. , vol.38 , pp. 145-162
    • Ingraham, M.H.1    Wegner, K.W.2
  • 22
    • 0042244904 scopus 로고
    • Paare symmetrischer und alternierender Formen zweiten Grades
    • W. Klingenberg Paare symmetrischer und alternierender Formen zweiten Grades Abh. Math. Sem. Univ. Hamburg 19 1954 78 93
    • (1954) Abh. Math. Sem. Univ. Hamburg , vol.19 , pp. 78-93
    • Klingenberg, W.1
  • 24
    • 0002616641 scopus 로고
    • Über die congruenten Transformationen der bilinearer Formen
    • Berlin
    • L. Kronecker, Über die congruenten Transformationen der bilinearer Formen, Monats. der Akademie der Wissenschaften, Berlin (1974) 397-447.
    • (1974) Monats. der Akademie der Wissenschaften , pp. 397-447
    • Kronecker, L.1
  • 26
    • 0040451231 scopus 로고
    • Invariant neutral subspaces for symmetric and skew-matrix pairs
    • P. Lancaster, and L. Rodman Invariant neutral subspaces for symmetric and skew-matrix pairs Canad. J. Math. 46 1994 602 618
    • (1994) Canad. J. Math. , vol.46 , pp. 602-618
    • Lancaster, P.1    Rodman, L.2
  • 27
    • 0039266531 scopus 로고
    • Minimal symmetric factorizations of symmetric real and complex rational matrix functions
    • P. Lancaster, and L. Rodman Minimal symmetric factorizations of symmetric real and complex rational matrix functions Linear Algebra Appl. 220 1995 249 282
    • (1995) Linear Algebra Appl. , vol.220 , pp. 249-282
    • Lancaster, P.1    Rodman, L.2
  • 28
    • 0041045180 scopus 로고
    • Existence and uniqueness theorems for algebraic Riccati equations
    • P. Lancaster, and L. Rodman Existence and uniqueness theorems for algebraic Riccati equations Int. J. Control 32 1980 285 309
    • (1980) Int. J. Control , vol.32 , pp. 285-309
    • Lancaster, P.1    Rodman, L.2
  • 29
    • 24644490536 scopus 로고    scopus 로고
    • Canonical forms for hermitian matrix pairs under strict equivalence and congruence
    • in press
    • P. Lancaster, L. Rodman, Canonical forms for hermitian matrix pairs under strict equivalence and congruence, SIAM Rev., in press.
    • SIAM Rev.
    • Lancaster, P.1    Rodman, L.2
  • 32
    • 0000105740 scopus 로고
    • Canonical forms for symplectic and Hamiltonian matrices
    • A.J. Laub, and K. Meyer Canonical forms for symplectic and Hamiltonian matrices Celest. Mech. 9 1974 213 238
    • (1974) Celest. Mech. , vol.9 , pp. 213-238
    • Laub, A.J.1    Meyer, K.2
  • 33
    • 0000902795 scopus 로고    scopus 로고
    • Canonical forms for Hamiltonian and symplectic matrices and pencils
    • W.W. Lin, V. Mehrmann, and H. Xu Canonical forms for Hamiltonian and symplectic matrices and pencils Linear Algebra Appl. 302/303 1999 469 533
    • (1999) Linear Algebra Appl. , vol.302-303 , pp. 469-533
    • Lin, W.W.1    Mehrmann, V.2    Xu, H.3
  • 36
    • 0033633270 scopus 로고    scopus 로고
    • Condensed forms for skew-Hamiltonian/Hamiltonian pencils
    • C. Mehl Condensed forms for skew-Hamiltonian/Hamiltonian pencils SIAM J. Matrix Anal. Appl. 21 1999 454 476
    • (1999) SIAM J. Matrix Anal. Appl. , vol.21 , pp. 454-476
    • Mehl, C.1
  • 37
    • 1142307789 scopus 로고    scopus 로고
    • Canonical forms for doubly structured matrices and pencils
    • C. Mehl, V. Mehrmann, and H. Xu Canonical forms for doubly structured matrices and pencils Electron. J. Linear Algebra 7 2000 112 151
    • (2000) Electron. J. Linear Algebra , vol.7 , pp. 112-151
    • Mehl, C.1    Mehrmann, V.2    Xu, H.3
  • 38
    • 0037555939 scopus 로고    scopus 로고
    • Structured Jordan canonical forms for structured matrices that are Hermitian, skew-Hermitian, or unitary with respect to an indefinite inner product
    • V. Mehrmann, and H. Xu Structured Jordan canonical forms for structured matrices that are Hermitian, skew-Hermitian, or unitary with respect to an indefinite inner product Electron. J. Linear Algebra 5 1999 67 103
    • (1999) Electron. J. Linear Algebra , vol.5 , pp. 67-103
    • Mehrmann, V.1    Xu, H.2
  • 39
    • 49049146869 scopus 로고
    • A Schur decomposition for Hamiltonian matrices
    • C. Paige, and C. Van Loan A Schur decomposition for Hamiltonian matrices Linear Algebra Appl. 41 1981 11 32
    • (1981) Linear Algebra Appl. , vol.41 , pp. 11-32
    • Paige, C.1    Van Loan, C.2
  • 40
    • 0000013596 scopus 로고
    • Stability of invariant Lagrangian subspaces. I. Topics in operator theory
    • A.C.M. Ran, L. Rodman, Stability of invariant Lagrangian subspaces. I. Topics in operator theory, Oper. Theor.: Adv. Appl. 32 (1988) 181-218.
    • (1988) Oper. Theor.: Adv. Appl. , vol.32 , pp. 181-218
    • Ran, A.C.M.1    Rodman, L.2
  • 41
    • 0010971208 scopus 로고
    • Stability of invariant Lagrangian subspaces. II
    • A.C.M. Ran, L. Rodman, Stability of invariant Lagrangian subspaces. II, Oper. Theor.: Adv. Appl. 40 (1989) 391-425.
    • (1989) Oper. Theor.: Adv. Appl. , vol.40 , pp. 391-425
    • Ran, A.C.M.1    Rodman, L.2
  • 42
    • 44949291177 scopus 로고
    • Stable invariant Lagrangian subspaces; Factorization of symmetric rational matrix functions and other applications
    • A.C.M. Ran, and L. Rodman Stable invariant Lagrangian subspaces; factorization of symmetric rational matrix functions and other applications Linear Algebra Appl. 137/138 1990 575 620
    • (1990) Linear Algebra Appl. , vol.137-138 , pp. 575-620
    • Ran, A.C.M.1    Rodman, L.2
  • 43
    • 1442319257 scopus 로고    scopus 로고
    • A class of robustness problems in matrix analysis
    • A.C.M. Ran, and L. Rodman A class of robustness problems in matrix analysis Oper. Theor.: Adv. Appl. 134 2002 337 389
    • (2002) Oper. Theor.: Adv. Appl. , vol.134 , pp. 337-389
    • Ran, A.C.M.1    Rodman, L.2
  • 44
    • 21144479109 scopus 로고
    • Direct complements of invariant Lagrangian subspaces and minimal factorizations of skew-symmetric rational matrix functions
    • A.C.M. Ran, L. Rodman, and J.E. Rubin Direct complements of invariant Lagrangian subspaces and minimal factorizations of skew-symmetric rational matrix functions Linear Algebra Appl. 180 1993 61 94
    • (1993) Linear Algebra Appl. , vol.180 , pp. 61-94
    • Ran, A.C.M.1    Rodman, L.2    Rubin, J.E.3
  • 45
    • 0020750544 scopus 로고
    • Geometry of the algebraic Riccati equation. I, II
    • M.A. Shayman Geometry of the algebraic Riccati equation. I, II SIAM J. Control Optim. 21 1983 375 394 395-409.
    • (1983) SIAM J. Control Optim. , vol.21 , pp. 375-394
    • Shayman, M.A.1
  • 47
    • 0034389246 scopus 로고    scopus 로고
    • Canonical matrices for linear matrix problems
    • V.V. Sergeichuk Canonical matrices for linear matrix problems Linear Algebra Appl. 317 2000 53 102
    • (2000) Linear Algebra Appl. , vol.317 , pp. 53-102
    • Sergeichuk, V.V.1
  • 48
    • 0001750429 scopus 로고
    • The characteristic polynomial of a principal subpencil of a hermitian matrix pencil
    • R.C. Thompson The characteristic polynomial of a principal subpencil of a hermitian matrix pencil Linear Algebra Appl. 14 1976 135 177
    • (1976) Linear Algebra Appl. , vol.14 , pp. 135-177
    • Thompson, R.C.1
  • 49
    • 0001625882 scopus 로고
    • Pencils of complex and real symmetric and skew-matrices
    • R.C. Thompson Pencils of complex and real symmetric and skew-matrices Linear Algebra Appl. 147 1991 323 371
    • (1991) Linear Algebra Appl. , vol.147 , pp. 323-371
    • Thompson, R.C.1
  • 50
    • 23144450224 scopus 로고
    • On the canonical form of a non-singular pencil of Hermitian matrices
    • G.R. Trott On the canonical form of a non-singular pencil of Hermitian matrices Amer. J. Math. 56 1934 359 371
    • (1934) Amer. J. Math. , vol.56 , pp. 359-371
    • Trott, G.R.1
  • 51
    • 84963088619 scopus 로고
    • On the equivalence of pencils of Hermitian forms
    • H.W. Turnbull On the equivalence of pencils of Hermitian forms Proc. London Math. Soc. 39 1935 232 248
    • (1935) Proc. London Math. Soc. , vol.39 , pp. 232-248
    • Turnbull, H.W.1
  • 53
    • 0348166183 scopus 로고
    • Pairs of symmetric and skew-matrices in an arbitrary field. i
    • K. Venkatachaliengar Pairs of symmetric and skew-matrices in an arbitrary field. I Proc. Indian Acad. Sci., Sect. A 22 1945 243 264
    • (1945) Proc. Indian Acad. Sci., Sect. A , vol.22 , pp. 243-264
    • Venkatachaliengar, K.1
  • 54
    • 2542470867 scopus 로고    scopus 로고
    • On Hamiltonian and symplectic Lanczos processes
    • D.S. Watkins On Hamiltonian and symplectic Lanczos processes Linear Algebra Appl. 385 2004 23 45
    • (2004) Linear Algebra Appl. , vol.385 , pp. 23-45
    • Watkins, D.S.1
  • 56
    • 23144465852 scopus 로고
    • The equivalence of pairs of Hermitian matrices
    • K.W. Wegner The equivalence of pairs of Hermitian matrices Trans. Amer. Math. Soc. 38 1935 145 162
    • (1935) Trans. Amer. Math. Soc. , vol.38 , pp. 145-162
    • Wegner, K.W.1
  • 58
    • 23144442204 scopus 로고
    • The equivalence of nonsingular pencils of Hermitian matrices in an arbitrary field
    • J. Williamson The equivalence of nonsingular pencils of Hermitian matrices in an arbitrary field Amer. J. Math. 57 1935 475 490
    • (1935) Amer. J. Math. , vol.57 , pp. 475-490
    • Williamson, J.1
  • 59
    • 0001316020 scopus 로고
    • On the algebraic problem concerning the normal forms of linear dynamical systems
    • J. Williamson On the algebraic problem concerning the normal forms of linear dynamical systems Amer. J. Math. 58 1936 141 163
    • (1936) Amer. J. Math. , vol.58 , pp. 141-163
    • Williamson, J.1
  • 60
    • 23144453903 scopus 로고
    • The conjunctive equivalence of pencils of Hermitian and anti-Hermitian matrices
    • J. Williamson The conjunctive equivalence of pencils of Hermitian and anti-Hermitian matrices Amer. J. Math. 59 1937 399 413
    • (1937) Amer. J. Math. , vol.59 , pp. 399-413
    • Williamson, J.1
  • 61
    • 23144442447 scopus 로고
    • Note on the equivalence of nonsingular pencils of Hermitian matrices
    • J. Williamson Note on the equivalence of nonsingular pencils of Hermitian matrices Bull. Amer. Math. Soc. 51 1945 894 897
    • (1945) Bull. Amer. Math. Soc. , vol.51 , pp. 894-897
    • Williamson, J.1
  • 62
    • 23144449552 scopus 로고
    • Quadratic and skew-symmetric bilinear forms in a real symplectic space
    • I.M. Yaglom Quadratic and skew-symmetric bilinear forms in a real symplectic space Trudy Sem. Vektor. Tenzor. Analizu 8 1950 364 381 (in Russian).
    • (1950) Trudy Sem. Vektor. Tenzor. Analizu , vol.8 , pp. 364-381
    • Yaglom, I.M.1


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