The goal of this note is to give a proof of the following result:
Theorem 1 Let be an odd dimensional vector space over and let be linear. Then admits an eigenvector.
The proof that follows is in the spirit of Axler’s textbook, so it avoids the use of determinants. However, I feel it is easier than the argument in the book, and it has the additional advantage of not depending on the fundamental theorem of algebra. In fact, the motivation for finding this argument was to avoid the use of the fundamental theorem.
The proof we present of Theorem 1 can be seen as an elaboration of the argument in the case when that we discussed in lecture. It was found by David Milovich in Facebook.
Proof: The argument is by induction on with the case being obvious.
Assume then that is odd, that the result holds for all positive odd dimensions smaller than and that is linear. We may also assume that the result is false for , and argue by contradiction. It follows that is invertible for all (where is the identity operator), and that if is a proper -invariant subspace, then is even.
Lemma 2 Any is contained in a proper -invariant subspace.
Proof: This is clear if Otherwise, recall that since are linearly dependent, there is a polynomial such that and is non-constant and of degree If we are done, because then is -invariant and of dimension at most If we use that has a real root, to factor for some and of degree Since is invertible, we have that and, since we are done.
Let be a maximal proper -invariant subspace. let and let be a proper -invariant subspace with Note that is also -invariant, and strictly larger than since it contains By maximality of we must have Since and have even dimension, it follows that is odd. Since is -invariant and proper, the inductive hypothesis gives us a contradiction, and we are done.
Now, combining this proof with the results in The fundamental theorem of algebra and linear algebra by Harm Derksen, American Mathematical Monthly, 110 (7) (2003), 620-623 (also available through JSTOR) or in the preprint The fundamental theorem of algebra via linear algebra by Keith Conrad, we have a proof of the existence of eigenvectors for operators on finite dimensional complex vector spaces that does not require the use of determinants and does not use the fundamental theorem of algebra. By means of a well-known trick (the observation before the statement of Theorem 2 in Conrad’s paper, or Corollary 8 in Derksen’s), we actually obtain (once determinants are introduced) the fundamental theorem of algebra as a corollary of this purely linear algebraic result. Of course, at the heart of this proof are the two facts that any odd degree polynomial with real coefficients has a real root (used in the proof of Theorem 1), and that any complex number admits a complex square root, which are needed in all proofs of the fundamental theorem of algebra.
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