Eigenvalues of symmetric integer matrices. (English) Zbl 0759.15007
This paper contains the proof of the following interesting theorem: The set of eigenvalues of symmetric matrices over the rational integer is the set of all totally real algebraic integers.
The original question was posed by A. Hoffman (unpublished communication) and the analogous one over the rational numbers was answered by F. Krakowski [Commentarii Math. Helvet. 32, 224-240 (1958; Zbl 0083.007)]. The proof of the theorem of this paper shows that a totally real algebraic integer \(\Theta\) occurs as an eigenvalue of a \(t\times t\) integer matrix with \(t=O(n^ 2)\) where \(n=[Q(\Theta):Q]\). The author also poses the question how to reduce the size of the considered matrix to \(O(n)\) similarly to the case discussed over the rational numbers.
The original question was posed by A. Hoffman (unpublished communication) and the analogous one over the rational numbers was answered by F. Krakowski [Commentarii Math. Helvet. 32, 224-240 (1958; Zbl 0083.007)]. The proof of the theorem of this paper shows that a totally real algebraic integer \(\Theta\) occurs as an eigenvalue of a \(t\times t\) integer matrix with \(t=O(n^ 2)\) where \(n=[Q(\Theta):Q]\). The author also poses the question how to reduce the size of the considered matrix to \(O(n)\) similarly to the case discussed over the rational numbers.
Reviewer: Á.G.Horváth (Budapest)
MSC:
| 15B36 | Matrices of integers |
| 11C20 | Matrices, determinants in number theory |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Citations:
Zbl 0083.007References:
| [1] | Bender, E. A., The dimensions of symmetric matrices with a given minimum polynomial, Linear Algebra Appl., 3, 115-123 (1970) · Zbl 0186.05601 |
| [2] | C. Delorme; C. Delorme · Zbl 0784.05045 |
| [3] | A. Hoffman; A. Hoffman |
| [4] | Krakowski, F., Eigenwerte und Minimalpolynome symmetrischer Matrizen in Kommutativen Körpern, Comment Math. Helv., 32, 224-240 (1958) · Zbl 0083.00701 |
| [5] | Lang, S., (Algebraic Number Theory (1986), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0601.12001 |
| [6] | O’Meara, O. T., (Introduction to Quadratic Forms (1963), Academic Press: Academic Press Orlando, FL) · Zbl 0107.03301 |
| [7] | Taussky, O., On matrix classes corresponding to an ideal and its inverse, Illinois J. Math., 1, 108-113 (1957) · Zbl 0078.02903 |
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