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1988•American Mathematical MonthlyRequires access

Companion Matrices with Integer Entries and Integer Eigenvalues and Eigenvectors

Richard C. Gilbert

Open publisher page 8 citations

Abstract

Recent articles in the MONTHLY have discussed the construction of matrices with integer entries, integer eigenvalues, and integer eigenvectors (see [3], [4], [6]). In this note we give a simple method for this construction by use of companion matrices [1, chap. X, ?5]. The method gives, in addition, matrices whose eigenvalues have any specified algebraic multiplicity but spectral multiplicity one. This is related to [2]. The corresponding Jordan strings have integer entries. Finally, the method provides matrices with integer entries whose inverses also have integer entries. Consider a monic polynomial of degree n,

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Recent articles in the MONTHLY have discussed the construction of matrices with integer entries, integer eigenvalues, and integer eigenvectors (see [3], [4], [6]). In this note we give a simple method for this construction by use of companion matrices [1, chap. X, ?5]. The method gives, in addition, matrices whose eigenvalues have any specified algebraic multiplicity but spectral multiplicity one. This is related to [2]. The corresponding Jordan strings have integer entries. Finally, the method provides matrices with integer entries whose inverses also have integer entries. Consider a monic polynomial of degree n,

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Available abstract

Recent articles in the MONTHLY have discussed the construction of matrices with integer entries, integer eigenvalues, and integer eigenvectors (see [3], [4], [6]). In this note we give a simple method for this construction by use of companion matrices [1, chap. X, ?5]. The method gives, in addition, matrices whose eigenvalues have any specified algebraic multiplicity but spectral multiplicity one. This is related to [2]. The corresponding Jordan strings have integer entries. Finally, the method provides matrices with integer entries whose inverses also have integer entries. Consider a monic polynomial of degree n,

Key concepts: Integer points in convex polyhedra, Integer (computer science), Eigenvalues and eigenvectors, Integer matrix, Mathematics, Radical of an integer, Multiplicity (mathematics), Combinatorics

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