Companion Matrices with Integer Entries and Integer Eigenvalues and Eigenvectors
Richard C. Gilbert · American Mathematical Monthly · 1988
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,