## Distinct Eigenvalues

Today I’d like to point out a little fact that applies over any field (not just the algebraically-closed ones). Let be a linear endomorphism on a vector space , and for , let be eigenvectors with corresponding eigenvalues . Further, assume that for . I claim that the are linearly independent.

Suppose the collection is linearly dependent. Then for some we have a linear relation

We can assume that is the smallest index so that we get such a relation involving only smaller indices.

Hit both sides of this equation by , and use the eigenvalue properties to find

On the other hand, we could just multiply the first equation by to get

Subtracting, we find the equation

But we this would contradict the minimality of we assumed before. Thus there can be no such linear relation, and eigenvectors corresponding to distinct eigenvalues are linearly independent.