Some matrix completion problems

Glória Cravo

doi:10.1016/j.crma.2007.04.016

Algebra

Some matrix completion problems

Presented by: Jean-Pierre Serre

Glória Cravo ¹

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

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

Abstract
Résumé

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.

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.

Received: 2006-05-15
Accepted: 2007-04-18
Published online: 2007-05-25

DOI: 10.1016/j.crma.2007.04.016

Author's affiliations:

Glória Cravo ¹

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

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     author = {Gl\'oria Cravo},
     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},
}

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Glória Cravo. Some matrix completion problems. Comptes Rendus. Mathématique, Volume 344 (2007) no. 11, pp. 673-676. doi : 10.1016/j.crma.2007.04.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.016/

References
Cited by

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

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

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

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

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