Some matrix completion problems

Cravo, Glória

doi:10.1016/j.crma.2007.04.016

Algebra

Some matrix completion problems
[Quelques problèmes de complétion de matrices]

Présenté par : Jean-Pierre Serre

Cravo, Glória ¹

¹ Departamento de Matemática e Engenharias, Universidade da Madeira, 9000-390 Funchal, Madeira, Portugal

Comptes Rendus. Mathématique, Tome 344 (2007) no. 11, pp. 673-676.

Résumé
Abstract

Soit F un corps et soient $n, p_{1}, p_{2}, p_{3}$ des entiers positifs tels que $n = p_{1} + p_{2} + p_{3}$ . Soit

A = [\begin{matrix} A_{1, 1} & A_{1, 2} & A_{1, 3} \\ A_{2, 1} & A_{2, 2} & A_{2, 3} \\ A_{3, 1} & A_{3, 2} & A_{3, 3} \end{matrix}] \in F^{n \times n},

où les blocs

A_{i, j}

sont de type

p_{i} \times p_{j}, i, j \in {1, 2, 3}

. Nous étudions le polynôme caractéristique de la matrice A, quand :

(i) $A_{1, 1}, A_{1, 2}$ et $A_{3, 3}$ sont connus et les autres blocs varient ;

(ii) $A_{1, 1}, A_{1, 2}$ et $A_{2, 3}$ sont connus et les autres blocs varient.

Let F be a field and let $n, p_{1}, p_{2}, p_{3}$ be positive integers such that $n = p_{1} + p_{2} + p_{3}$ . Let

A = [\begin{matrix} A_{1, 1} & A_{1, 2} & A_{1, 3} \\ A_{2, 1} & A_{2, 2} & A_{2, 3} \\ A_{3, 1} & A_{3, 2} & A_{3, 3} \end{matrix}] \in F^{n \times n},

where the blocks

A_{i, j}

are of type

p_{i} \times p_{j}, i, j \in {1, 2, 3}

, and the blocks in the positions

(i, i)

are square. We describe the characteristic polynomial of A, when:

(i) $A_{1, 1}$ , $A_{1, 2}$ and $A_{3, 3}$ are fixed and the other blocks vary;

(ii) $A_{1, 1}$ , $A_{1, 2}$ and $A_{2, 3}$ are fixed and the remaining blocks vary.

Reçu le : 2006-05-15
Accepté le : 2007-04-18
Publié le : 2007-05-25

DOI : 10.1016/j.crma.2007.04.016

Affiliations des auteurs :

Cravo, Glória ¹

¹ Departamento de Matemática e Engenharias, Universidade da Madeira, 9000-390 Funchal, Madeira, Portugal

@article{CRMATH_2007__344_11_673_0,
     author = {Cravo, Gl\'oria},
     title = {Some matrix completion problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {673--676},
     publisher = {Elsevier},
     volume = {344},
     number = {11},
     year = {2007},
     doi = {10.1016/j.crma.2007.04.016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.04.016/}
}

TY  - JOUR
AU  - Cravo, Glória
TI  - Some matrix completion problems
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 673
EP  - 676
VL  - 344
IS  - 11
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.04.016/
DO  - 10.1016/j.crma.2007.04.016
LA  - en
ID  - CRMATH_2007__344_11_673_0
ER  -

%0 Journal Article
%A Cravo, Glória
%T Some matrix completion problems
%J Comptes Rendus. Mathématique
%D 2007
%P 673-676
%V 344
%N 11
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.04.016/
%R 10.1016/j.crma.2007.04.016
%G en
%F CRMATH_2007__344_11_673_0

Cravo, Glória. Some matrix completion problems. Comptes Rendus. Mathématique, Tome 344 (2007) no. 11, pp. 673-676. doi : 10.1016/j.crma.2007.04.016. http://www.numdam.org/articles/10.1016/j.crma.2007.04.016/

Bibliographie
Cité par

[1] G. Cravo, Eigenvalues of matrices with prescribed submatrices, A. D. Journal of Mathematics, in press

[2] G. Cravo, Matrices with prescribed eigenvalues and prescribed submatrices, Preprint

[3] G. Cravo, Valores Próprios de Matrizes Parcialmente Conhecidas, PhD thesis, University of Lisbon, 2002

[4] Hershkowitz, D. Existence of matrices with prescribed eigenvalues and entries, Linear and Multilinear Algebra, Volume 14 (1983), pp. 315-342

[5] Oliveira, G.N. Matrices with prescribed characteristic polynomial and several prescribed submatrices, Linear and Multilinear Algebra, Volume 2 (1975), pp. 357-364

[6] Sá, E.M. Imbedding conditions for λ-matrices, Linear Algebra Appl., Volume 24 (1979), pp. 33-50

[7] Thompson, R.C. Interlacing inequalities for invariant factors, Linear Algebra Appl., Volume 24 (1979), pp. 1-31

Cité par Sources :