16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.1.7
The Rational and Jordan Forms
Mathematics — Linear Algebra · diagonalizable-operators-cyclic-vectors
Problem
Let V be an n-dimensional vector space, and let T be a linear operator on V. Suppose that T is diagonalizable.
(a) If T has a cyclic vector, show that T has n distinct characteristic values.
(b) If T has n distinct characteristic values, and if {α1,…,αn} is a basis of characteristic vectors for T, show that α=α1+⋯+αn is a cyclic vector for T.