Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality
Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that t...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Kolotilina, L. Yu [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1998 |
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| Schlagwörter: |
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| Anmerkung: |
© Plenum Publishing Corporation 1998 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of mathematical sciences - Kluwer Academic Publishers-Plenum Publishers, 1994, 89(1998), 6 vom: Mai, Seite 1690-1693 |
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| Übergeordnetes Werk: |
volume:89 ; year:1998 ; number:6 ; month:05 ; pages:1690-1693 |
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| DOI / URN: |
10.1007/BF02355372 |
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| 520 | |a Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. | ||
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10.1007/BF02355372 doi (DE-627)OLC2030750735 (DE-He213)BF02355372-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 31.00 bkl Kolotilina, L. Yu verfasserin aut Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality Enthalten in Journal of mathematical sciences Kluwer Academic Publishers-Plenum Publishers, 1994 89(1998), 6 vom: Mai, Seite 1690-1693 (DE-627)18219762X (DE-600)1185490-X (DE-576)038888130 1072-3374 nnns volume:89 year:1998 number:6 month:05 pages:1690-1693 https://doi.org/10.1007/BF02355372 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4325 31.00 VZ AR 89 1998 6 05 1690-1693 |
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10.1007/BF02355372 doi (DE-627)OLC2030750735 (DE-He213)BF02355372-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 31.00 bkl Kolotilina, L. Yu verfasserin aut Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality Enthalten in Journal of mathematical sciences Kluwer Academic Publishers-Plenum Publishers, 1994 89(1998), 6 vom: Mai, Seite 1690-1693 (DE-627)18219762X (DE-600)1185490-X (DE-576)038888130 1072-3374 nnns volume:89 year:1998 number:6 month:05 pages:1690-1693 https://doi.org/10.1007/BF02355372 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4325 31.00 VZ AR 89 1998 6 05 1690-1693 |
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10.1007/BF02355372 doi (DE-627)OLC2030750735 (DE-He213)BF02355372-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 31.00 bkl Kolotilina, L. Yu verfasserin aut Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality Enthalten in Journal of mathematical sciences Kluwer Academic Publishers-Plenum Publishers, 1994 89(1998), 6 vom: Mai, Seite 1690-1693 (DE-627)18219762X (DE-600)1185490-X (DE-576)038888130 1072-3374 nnns volume:89 year:1998 number:6 month:05 pages:1690-1693 https://doi.org/10.1007/BF02355372 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4325 31.00 VZ AR 89 1998 6 05 1690-1693 |
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10.1007/BF02355372 doi (DE-627)OLC2030750735 (DE-He213)BF02355372-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 31.00 bkl Kolotilina, L. Yu verfasserin aut Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality Enthalten in Journal of mathematical sciences Kluwer Academic Publishers-Plenum Publishers, 1994 89(1998), 6 vom: Mai, Seite 1690-1693 (DE-627)18219762X (DE-600)1185490-X (DE-576)038888130 1072-3374 nnns volume:89 year:1998 number:6 month:05 pages:1690-1693 https://doi.org/10.1007/BF02355372 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4325 31.00 VZ AR 89 1998 6 05 1690-1693 |
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10.1007/BF02355372 doi (DE-627)OLC2030750735 (DE-He213)BF02355372-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 31.00 bkl Kolotilina, L. Yu verfasserin aut Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. Monotone Function Diagonal Entry Hermitian Matrix Hermitian Matrice Block Diagonality Enthalten in Journal of mathematical sciences Kluwer Academic Publishers-Plenum Publishers, 1994 89(1998), 6 vom: Mai, Seite 1690-1693 (DE-627)18219762X (DE-600)1185490-X (DE-576)038888130 1072-3374 nnns volume:89 year:1998 number:6 month:05 pages:1690-1693 https://doi.org/10.1007/BF02355372 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_22 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4325 31.00 VZ AR 89 1998 6 05 1690-1693 |
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interrelations between eigenvalues and diagonal entries of hermitian matrices implying their block diagonality |
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Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality |
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Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. © Plenum Publishing Corporation 1998 |
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Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. © Plenum Publishing Corporation 1998 |
| abstract_unstemmed |
Abstract Let A=($ a_{ij} $)i,jn=1 be a Hermitian matrix and let$$\lambda _1 \geqslant \lambda _2 \geqslant \ldots \geqslant \lambda _n $$ denote its eigenvalues. If$$\sum\limits_{i = 1}^k {\lambda _i } = \sum\limits_{i = 1}^k {a_{ii} } $$, k<n, then A is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function f(t), and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles. © Plenum Publishing Corporation 1998 |
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Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality |
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