## Mathematics, Department of

#### Date of this Version

1930

#### Abstract

One of Sylvester's theorems f on matrices states that if the characteristic equation

(1) | *M* - λ*I*| = f(λ) = 0

of a square matrix *M* has the roots λ_{1}, λ_{2}, … , λ_{n}, then the characteristic equation

(2) | φ*M* - ρ*I*| = = *g*(ρ) = 0

of any integral function of *M*, namely, φ*M*, 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.

## Comments

Published in

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