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## Schur-like forms for matrix Lie groups, Lie algebras and Jordan algebras (1997)

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Citations: | 15 - 6 self |

### Citations

1768 |
Differential Geometry, Lie Groups and Symmetric Spaces
- Helgason
- 1978
(Show Context)
Citation Context ... the skew-Hermitian form is represented by the matrix J = 0 I n \GammaI n 0 ; where m = 2n. The classical Lie groups we consider here are the matrices that are unitary with respect to J or \Sigma p;q =-=[15]-=-. We will discuss only classes of complex matrices here, but analogous results also exist for the real case. Definition 1 1) The Lie group O p;q of \Sigma p;q -unitary matrices is defined by O p;q = f... |

1477 |
The Algebraic Eigenvalue Problem
- Wilkinson
- 1965
(Show Context)
Citation Context ... Hermitian or unitary, then the matrix is normal, and the Jordan form and the Schur form coincide. Consequently, complete eigenstructure information can be obtained via a numerically stable procedure =-=[14, 26, 29]-=- for matrices in these classes. We may expect that between the general case and these special cases there are more refined condensed forms for matrices from the classes defined above. For the classes ... |

981 |
The Symmetric Eigenvalue Problem
- Parlett
- 1980
(Show Context)
Citation Context ...e F G H F H : F; G; H 2 C n;n ; G = \GammaG H ; H = \GammaH H oe ; is the Jordan algebra of J-skew Hermitian matrices or skew Hamiltonian matrices. For applications of these classes see, for example, =-=[5, 26, 27]-=-. In this paper we discuss structure-preserving similarity transformations to condensed forms from which the eigenvalues of the matrices can be read off in a simple way. For general matrices these are... |

949 |
The Theory of Matrices,
- Gantmacher
- 1990
(Show Context)
Citation Context ... which the eigenvalues of the matrices can be read off in a simple way. For general matrices these are the Jordan canonical form (under similarity transformations with nonsingular matrices), see e.g. =-=[13]-=-, and the Schur form (under similarity with unitary matrices), see e.g. [14]. While both the Jordan form and Schur form display all the eigenvalues, the transformation to Jordan form gives the eigenve... |

326 |
Algebraic Riccati Equations
- Lancaster
- 1995
(Show Context)
Citation Context ...GammaF H : F; G; H 2 C n;n ; G = G H ; H = H H oe : These Lie algebras also have great importance in practical applications, see [4, 5, 25] for applications of \Sigma p;q -skew symmetric matrices and =-=[8, 23, 20]-=- for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algebras. These are the Jordan algebras [3, 16] associated with the two Lie g... |

203 |
The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Number 163
- Mehrmann
- 1991
(Show Context)
Citation Context ...defined by Sp 2n = fG 2 C 2n;2n : G H JG = Jg. Developing structure-preserving numerical methods for solving eigenvalue problems for matrices in these groups remains an active area of recent research =-=[7, 9, 10, 23]-=-, motivated by applications arising in signal processing [1] and optimal control for discrete-time or continuous-time linear systems, see [23] and the references therein. Of equal importance are the L... |

198 |
Structure and Representations of Jordan Algebras
- Jacobson
- 1968
(Show Context)
Citation Context ...ew symmetric matrices and [8, 23, 20] for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algebras. These are the Jordan algebras =-=[3, 16] asso-=-ciated with the two Lie groups. Definition 3 1") C p;q =fC 2 C p+q;p+q : \Sigma p;q C \Gamma C H \Sigma p;q = 0g = ae F G \GammaG H H : F = F H 2 C p;p ; H = H H 2 C q;q ; G 2 C p;q oe ; is the J... |

178 |
Geometric Algebra, Interscience
- Artin
- 1957
(Show Context)
Citation Context ...otes the conjugate transpose of the column vector x. We restrict ourselves to the case that ! x; y ?= 0 if and only if ! y; x ?= 0: This condition implies that K is either Hermitian or skew-Hermitian =-=[2]-=-. If K is Hermitian, we can perform a change of basis on C m so that the sesquilinear form is represented by the matrix \Sigma p;q = \GammaI p 0 0 I q ; where p + q = m, ps0, and qs0. On the other han... |

105 |
Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys
- Kogut, Susskind
- 1975
(Show Context)
Citation Context ...apply. For case iv) the computation of the eigenvalues reduces to the computation of the singular values of the matrix \Theta C H 1 C H 2 B . An important special case that arises in particle physics =-=[18, 19]-=-, is the case p = q and ~ p = 0. In this case the matrices have the forms0 C \GammaC H 0 ; (27) where C 2 C p\Thetap . Again, the eigenvalues can be determined via the singular values of C. 5.2 Inters... |

56 |
A Symplectic Method for Approximating all the Eigenvalues of a Hamiltonian Matrix
- Loan
- 1984
(Show Context)
Citation Context ...e F G H F H : F; G; H 2 C n;n ; G = \GammaG H ; H = \GammaH H oe ; is the Jordan algebra of J-skew Hermitian matrices or skew Hamiltonian matrices. For applications of these classes see, for example, =-=[5, 26, 27]-=-. In this paper we discuss structure-preserving similarity transformations to condensed forms from which the eigenvalues of the matrices can be read off in a simple way. For general matrices these are... |

54 | Canonical forms for Hamiltonian and symplectic matrices and pencils
- Lin, Mehrmann, et al.
- 1999
(Show Context)
Citation Context ...rresponding to λ satisfies (3). Proof. This result was first stated and proved in [22]. A simpler proof based on canonical forms under symplectic similarity transformations has recently been given in =-=[24]-=-. Note that there are also more refined Jordan-like forms for matrices in H2n,SH2n and Sp2n, which do not have a triangular structure, see [21]. It follows from a result in [6] that the symplectic mat... |

44 |
Matrix factorizations for symplectic QR-like methods
- Bunse-Gerstner
- 1986
(Show Context)
Citation Context ... and proved in [22]. Note that there are also more refined Jordan-like forms for matrices in H 2n ; SH 2n and Sp 2n , which do not have a triangular structure, see [21]. 6 It follows from a result in =-=[6]-=- that the symplectic matrices Q in each part of Theorem 7 can be chosen to be unitary symplectic. However, matrices in the J classes exist for which the forms (2)--(5) can be achieved only via nonsymp... |

36 |
An analysis of the HR algorithm for computing the eigenvalues of a matrix
- Bunse-Gerstner
- 1981
(Show Context)
Citation Context ...atrices) is defined by H 2n =fA 2 C 2n;2n : JA +A H J = 0g = ae F G H \GammaF H : F; G; H 2 C n;n ; G = G H ; H = H H oe : These Lie algebras also have great importance in practical applications, see =-=[4, 5, 25]-=- for applications of \Sigma p;q -skew symmetric matrices and [8, 23, 20] for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algeb... |

34 |
A chart of numerical methods for structured eigenvalue problems
- Bunse-Gerstner, Byers, et al.
- 1992
(Show Context)
Citation Context ...GammaF H : F; G; H 2 C n;n ; G = G H ; H = H H oe : These Lie algebras also have great importance in practical applications, see [4, 5, 25] for applications of \Sigma p;q -skew symmetric matrices and =-=[8, 23, 20]-=- for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algebras. These are the Jordan algebras [3, 16] associated with the two Lie g... |

33 |
Normal forms of elements of classical real and complex Lie and Jordan algebras
- Djoković, Patera, et al.
- 1983
(Show Context)
Citation Context ...spaces. However, the numerical computation of the Schur form is 3 a well-conditioned problem, while the reduction to Jordan canonical form is in general an ill-conditioned problem, see e.g. [14]. See =-=[28]-=- for classifications of the structured Jordan forms for the classes of matrices defined above. The Jordan structure of a matrix can be computed, with considerably more effort than computing the Schur ... |

23 |
An algorithm for the numerical computation of the Jordan normal form of a complex matrix
- Kågström, Ruhe
- 1980
(Show Context)
Citation Context ...ure of a matrix can be computed, with considerably more effort than computing the Schur form, by computing the Wyer characteristics, which are invariants under unitary similarity transformations, see =-=[17]-=-. But if the matrix has an extra symmetry structure, for example if it is Hermitian, skew Hermitian or unitary, then the matrix is normal, and the Jordan form and the Schur form coincide. Consequently... |

23 |
Canonical forms for symplectic and Hamiltonian matrices
- Laub, Meyer
- 1974
(Show Context)
Citation Context ... Proof. This result was first stated and proved in [22]. Note that there are also more refined Jordan-like forms for matrices in H 2n ; SH 2n and Sp 2n , which do not have a triangular structure, see =-=[21]-=-. 6 It follows from a result in [6] that the symplectic matrices Q in each part of Theorem 7 can be chosen to be unitary symplectic. However, matrices in the J classes exist for which the forms (2)--(... |

15 |
An eigenvalue algorithm for skew-symmetric matrices
- PAARDEKOOPER
- 1971
(Show Context)
Citation Context ...atrices) is defined by H 2n =fA 2 C 2n;2n : JA +A H J = 0g = ae F G H \GammaF H : F; G; H 2 C n;n ; G = G H ; H = H H oe : These Lie algebras also have great importance in practical applications, see =-=[4, 5, 25]-=- for applications of \Sigma p;q -skew symmetric matrices and [8, 23, 20] for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algeb... |

14 |
Determination of Pisarenko frequency estimates as eigenvalues of an orthogonal matrix
- Ammar, Gragg, et al.
- 1987
(Show Context)
Citation Context ...serving numerical methods for solving eigenvalue problems for matrices in these groups remains an active area of recent research [7, 9, 10, 23], motivated by applications arising in signal processing =-=[1]-=- and optimal control for discrete-time or continuous-time linear systems, see [23] and the references therein. Of equal importance are the Lie algebras A p;q and H 2n corresponding to the Lie groups O... |

14 |
On some algebraic problems in connection with general eigenvalue algorithms
- Elsner
- 1979
(Show Context)
Citation Context ...defined by Sp 2n = fG 2 C 2n;2n : G H JG = Jg. Developing structure-preserving numerical methods for solving eigenvalue problems for matrices in these groups remains an active area of recent research =-=[7, 9, 10, 23]-=-, motivated by applications arising in signal processing [1] and optimal control for discrete-time or continuous-time linear systems, see [23] and the references therein. Of equal importance are the L... |

14 |
Solution of large matrix equations which occur in response theory
- Olson, Jensen, et al.
- 1988
(Show Context)
Citation Context ...vated our interest in analyzing multi-structured matrices. Matrices from the set H 2n " C n;n = ae A B \GammaB \GammaA : A = A H ; B = B H oe (35) arise in linear response theory in quantum chemi=-=stry [11, 12, 24]. By -=-definition, similarity transformations with matrices from O n;n " Sp 2n will preserve the structure. The elements of H 2n have the eigenvalue-symmetry '; \Gamma' while we find the eigenvalue-symm... |

13 |
On Schur type decompositions for Hamiltonian and symplectic pencils
- Lin, Ho
- 1990
(Show Context)
Citation Context ...luesof M has even algebraic multiplicity, say 2k, and any basis X k 2 C 2n;2k of the maximal invariant subspace for M corresponding tossatisfies (3). Proof. This result was first stated and proved in =-=[22]-=-. Note that there are also more refined Jordan-like forms for matrices in H 2n ; SH 2n and Sp 2n , which do not have a triangular structure, see [21]. 6 It follows from a result in [6] that the symple... |

7 |
A kqz algorithm for solving linear-response eigenvalue equations
- Flaschka, Lin, et al.
- 1992
(Show Context)
Citation Context ...ondensed forms for matrices which lie in the intersection of two of the classes defined above. Our main motivation to work on this topic arose from a class of matrices that occur in quantum chemistry =-=[11, 12, 24]. In -=-linear response theory one needs to compute eigenvalues and eigenvectors of matrices of the form A B \GammaB \GammaA ; A; B 2 C n;n ; A = A H ; B = B H : (1) Such matrices are clearly in H 2n "C ... |

5 |
Eine Variante des Lanczos-Algorithmus fur groe, dunn besetzte symmetrische Matrizen mit Blockstruktur. Dissertation, Universitat
- Flaschka
- 1992
(Show Context)
Citation Context ...ondensed forms for matrices which lie in the intersection of two of the classes defined above. Our main motivation to work on this topic arose from a class of matrices that occur in quantum chemistry =-=[11, 12, 24]. In -=-linear response theory one needs to compute eigenvalues and eigenvectors of matrices of the form A B \GammaB \GammaA ; A; B 2 C n;n ; A = A H ; B = B H : (1) Such matrices are clearly in H 2n "C ... |

2 |
Numerical simulation in particle physics
- Karsch, Laermann
- 1993
(Show Context)
Citation Context ...apply. For case iv) the computation of the eigenvalues reduces to the computation of the singular values of the matrix \Theta C H 1 C H 2 B . An important special case that arises in particle physics =-=[18, 19]-=-, is the case p = q and ~ p = 0. In this case the matrices have the forms0 C \GammaC H 0 ; (27) where C 2 C p\Thetap . Again, the eigenvalues can be determined via the singular values of C. 5.2 Inters... |

1 |
HR{Algorithmus zur numerischen Bestimmung der Eigenwerte einer Matrix. Dissertation, Universitat
- Der
- 1978
(Show Context)
Citation Context ...atrices) is defined by H 2n =fA 2 C 2n;2n : JA +A H J = 0g = ae F G H \GammaF H : F; G; H 2 C n;n ; G = G H ; H = H H oe : These Lie algebras also have great importance in practical applications, see =-=[4, 5, 25]-=- for applications of \Sigma p;q -skew symmetric matrices and [8, 23, 20] for applications of Hamiltonian matrices. The third class of matrices we consider plays a role similar to that of the Lie algeb... |

1 |
Numerical solution for algebraic Riccati equations
- Bunse-Gerstner, Byers, et al.
- 1989
(Show Context)
Citation Context ...defined by Sp 2n = fG 2 C 2n;2n : G H JG = Jg. Developing structure-preserving numerical methods for solving eigenvalue problems for matrices in these groups remains an active area of recent research =-=[7, 9, 10, 23]-=-, motivated by applications arising in signal processing [1] and optimal control for discrete-time or continuous-time linear systems, see [23] and the references therein. Of equal importance are the L... |

1 |
Neuere Verfahren zur Bestimmung der Eigenwerte von Matrizen
- Elsner
- 1979
(Show Context)
Citation Context |