A matrix completion problem over integral domains: the case with 2n — 3 prescribed blocks

  • Alberto Borobia UNED.
  • Roberto Canogar UNED.
  • Helena Smigoc University College Dublin.
Palabras clave: Matrix completions, inverse eigenvalue problems, matrices over integral domains, completación de matrices, problemas de valor propio inverso, matrices sobre dominios integrales.

Resumen

Let ∧ = {λ1,...,λnk} be amultisetofelements ofanintegral domain R.Let P be a partially prescribed n X n block matrix such that each prescribed entry is a k—block (a k X k matrix over R). If P has at most 2n — 3 prescribed entries then the unprescribed entries of P can be filled with k—blocks to obtain a matrix over R with spectrum ∧ (some natural conditions on the prescribed entries are required). We describe an algorithm to construct such completion.

Biografía del autor

Alberto Borobia, UNED.
Departamento Matemáticas.
Roberto Canogar, UNED.
Departamento Matemáticas.
Helena Smigoc, University College Dublin.
School of Mathematical Sciences.

Citas

[1] A. Borobia, Inverse eigenvalue problems, in: Leslie Hogben (Ed.), Handbook of linear algebra, 2nd edition, Discrete Mathematics and its Applications, Chapman and Hall/CRC, to appear (chapter 28).

[2] A. Borobia and R. Canogar. Matrix completion problem over integral domains: the case with a diagonal of prescribed blocks. Lin. Alg. Appl., 436(1): pp. 222—236, (2012).

[3] A. Borobia, R. Canogar, and H. Smigoc. A matrix completion problem over integral domains: the case with 2n-3 prescribed entries. Lin. Alg. Appl., 433: pp. 606—617, (2010).

[4] Moody T. Chu, Fasma Diele, and Ivonne Sgura. Gradient flow methods for matrix completion with prescribed eigenvalues. Linear Algebra Appl., 379: pp. 85—112, (2004). Tenth Conference of the International Linear Algebra Society.

[5] Moody T. Chu and Gene H. Golub. Inverse eigenvalue problems: theory, algorithms, and applications. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, (2005).

[6] G. Cravo and F.C. Silva. Eigenvalues of matrices with several prescribed blocks. Lin. Alg. Appl., 311: pp. 13—24, (2000).

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

[8] Kh.D. Ikramov and V.N. Chugunov. Inverse matrix eigenvalue problems. J. Math. Sci. (New York), 98(1): pp. 51—136, (2000).

[9] Helena Smigoc. The inverse eigenvalue problem for nonnegative matrices. Linear Algebra Appl., 393: pp. 365—374, (2004).
Publicado
2017-03-23
Cómo citar
Borobia, A., Canogar, R., & Smigoc, H. (2017). A matrix completion problem over integral domains: the case with 2n — 3 prescribed blocks. Proyecciones. Journal of Mathematics, 33(2), 215-233. https://doi.org/10.4067/S0716-09172014000200007
Sección
Artículos