16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.4.2

The Rational and Jordan Forms

Mathematics — Linear Algebra · cyclic-vectors-invariant-factors

Problem

Let

T

be a linear operator on a finite-dimensional vector space and let

A

be the matrix of

T

in some ordered basis. Then

T

has a cyclic vector if and only if the determinants of the

(n-1) \times (n-1)

submatrices of

xI - A

are relatively prime.