A sin2Θ theorem for graded indefinite Hermitian matrices
This paper gives double angle theorems that bound the change in an invariant subspace of an indefinite Hermitian matrix in the graded form H=D*AD subject to a perturbation H→H=D*(A+ΔA)D. These theorems extend recent results on a definite Hermitian matrix in the graded form (Linear Algebra Appl. 311...
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| Published in | Linear algebra and its applications Vol. 359; no. 1-3; pp. 263 - 276 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.01.2003
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0024-3795 1873-1856 |
| DOI | 10.1016/S0024-3795(02)00424-X |
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| Abstract | This paper gives double angle theorems that bound the change in an invariant subspace of an indefinite Hermitian matrix in the graded form H=D*AD subject to a perturbation H→H=D*(A+ΔA)D. These theorems extend recent results on a definite Hermitian matrix in the graded form (Linear Algebra Appl. 311 (2000) 45) but the bounds here are more complicated in that they depend on not only relative gaps and norms of ΔA as in the definite case but also norms of some J-unitary matrices, where J is diagonal with ±1 on its diagonal. For two special but interesting cases, bounds on these J-unitary matrices are obtained to show that their norms are of moderate magnitude. |
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| AbstractList | This paper gives double angle theorems that bound the change in an invariant subspace of an indefinite Hermitian matrix in the graded form H=D*AD subject to a perturbation H→H=D*(A+ΔA)D. These theorems extend recent results on a definite Hermitian matrix in the graded form (Linear Algebra Appl. 311 (2000) 45) but the bounds here are more complicated in that they depend on not only relative gaps and norms of ΔA as in the definite case but also norms of some J-unitary matrices, where J is diagonal with ±1 on its diagonal. For two special but interesting cases, bounds on these J-unitary matrices are obtained to show that their norms are of moderate magnitude. |
| Author | Li, Ren-Cang Truhar, Ninoslav |
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| Cites_doi | 10.1109/78.134396 10.1016/S0024-3795(99)00198-6 10.1016/S0024-3795(99)00126-3 10.1016/0024-3795(94)00197-9 10.1016/S0024-3795(00)00077-X 10.1137/0707001 |
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| Keywords | Invariant subspaces Relative perturbation bounds |
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| References | Horn, Johnson (BIB2) 1990 Stewart, Sun (BIB7) 1990 Zha (BIB9) 1996; 240 Slapničar, Truhar (BIB6) 2000; 309 Truhar, Slapničar (BIB8) 1999; 301 Horn, Johnson (BIB3) 1991 Onn, Steinhardt, Bojanczyk (BIB5) 1991 Davis, Kahan (BIB1) 1970; 7 Li (BIB4) 2000; 311 Truhar (10.1016/S0024-3795(02)00424-X_BIB8) 1999; 301 Li (10.1016/S0024-3795(02)00424-X_BIB4) 2000; 311 Horn (10.1016/S0024-3795(02)00424-X_BIB3) 1991 Onn (10.1016/S0024-3795(02)00424-X_BIB5) 1991 Slapničar (10.1016/S0024-3795(02)00424-X_BIB6) 2000; 309 Zha (10.1016/S0024-3795(02)00424-X_BIB9) 1996; 240 Davis (10.1016/S0024-3795(02)00424-X_BIB1) 1970; 7 Stewart (10.1016/S0024-3795(02)00424-X_BIB7) 1990 Horn (10.1016/S0024-3795(02)00424-X_BIB2) 1990 |
| References_xml | – start-page: 1575 year: 1991 end-page: 1588 ident: BIB5 article-title: Hyperbolic singular value decompositions and applications publication-title: IEEE Trans. Acoustics, Speech, Signal Process. – year: 1990 ident: BIB2 article-title: Matrix Analysis – volume: 301 start-page: 171 year: 1999 end-page: 185 ident: BIB8 article-title: Relative perturbation bound for invariant subspaces of Hermitian matrices publication-title: Linear Algebra Appl. – volume: 7 start-page: 1 year: 1970 end-page: 46 ident: BIB1 article-title: The rotation of eigenvectors by a perturbation III publication-title: SIAM J. Numer. Anal. – volume: 309 start-page: 57 year: 2000 end-page: 72 ident: BIB6 article-title: Relative perturbation theory for hyperbolic eigenvalue problem publication-title: Linear Algebra Appl. – year: 1990 ident: BIB7 article-title: Matrix Perturbation Theory – volume: 311 start-page: 45 year: 2000 end-page: 60 ident: BIB4 article-title: Relative perturbation theory: (iv) sin2Θ theorems publication-title: Linear Algebra Appl. – volume: 240 start-page: 199 year: 1996 end-page: 205 ident: BIB9 article-title: A note on the existence of the hyperbolic singular value decomposition publication-title: Linear Algebra Appl. – year: 1991 ident: BIB3 article-title: Topics in Matrix Analysis – start-page: 1575 year: 1991 ident: 10.1016/S0024-3795(02)00424-X_BIB5 article-title: Hyperbolic singular value decompositions and applications publication-title: IEEE Trans. Acoustics, Speech, Signal Process. doi: 10.1109/78.134396 – year: 1990 ident: 10.1016/S0024-3795(02)00424-X_BIB2 – volume: 301 start-page: 171 year: 1999 ident: 10.1016/S0024-3795(02)00424-X_BIB8 article-title: Relative perturbation bound for invariant subspaces of Hermitian matrices publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(99)00198-6 – year: 1990 ident: 10.1016/S0024-3795(02)00424-X_BIB7 – volume: 309 start-page: 57 year: 2000 ident: 10.1016/S0024-3795(02)00424-X_BIB6 article-title: Relative perturbation theory for hyperbolic eigenvalue problem publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(99)00126-3 – volume: 240 start-page: 199 year: 1996 ident: 10.1016/S0024-3795(02)00424-X_BIB9 article-title: A note on the existence of the hyperbolic singular value decomposition publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)00197-9 – volume: 311 start-page: 45 year: 2000 ident: 10.1016/S0024-3795(02)00424-X_BIB4 article-title: Relative perturbation theory: (iv) sin2Θ theorems publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(00)00077-X – volume: 7 start-page: 1 year: 1970 ident: 10.1016/S0024-3795(02)00424-X_BIB1 article-title: The rotation of eigenvectors by a perturbation III publication-title: SIAM J. Numer. Anal. doi: 10.1137/0707001 – year: 1991 ident: 10.1016/S0024-3795(02)00424-X_BIB3 |
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| Title | A sin2Θ theorem for graded indefinite Hermitian matrices |
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