Matrices Whose Characteristic Equations are Cyclic

Authors

T. A. Pierce, University of Nebraska - Lincoln

Date of this Version

1930

Comments

Published in Bull. Amer. Math. Soc. 36 (1930) 262-264. Used by permission.

Abstract

One of Sylvester's theorems f on matrices states that if the characteristic equation
(1) | M - λI| = f(λ) = 0
of a square matrix M has the roots λ₁, λ₂, … , λ_n, then the characteristic equation
(2) | φM - ρI| = = g(ρ) = 0
of any integral function of M, namely, φM, has the roots ρ_i = φ (λ_i), i = 1, 2, … , n. In this note an isomorphism is shown to exist between the algebraic and matric roots of (1) when this equation is cyclic. Certain consequences of this isomorphism are given.