Let S, T be commuting F-linear endomorphisms of a (nonzero) finite dimensional vector space over an algebraically closed field F. Suppose that S has two distinct eigenvalues. Show that S and T have at least two common linearly independent eigenvectors.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 26EQ
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Let S, T be commuting F-linear endomorphisms of a (nonzero) finite dimensional vector space over an algebraically closed field F. Suppose that S has two distinct eigenvalues. Show that S and T have at least two common linearly independent eigenvectors.

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