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Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue

Statement

Let $V$ be a nonzero finite-dimensional vector space over an algebraically closed field $F$ . Every endomorphism $T : V \to V$ has an eigenvalue in $F$ .

Facts & Assumptions

Given: A nonzero finite-dimensional $F$ -vector space $V$ , an algebraically closed field $F$ , and $T \in L (V)$ .

[L1]

Every nonconstant polynomial over an algebraically closed field has a root in that field (An algebraically closed field: every nonconstant polynomial has a root in the field).

[L2]

The characteristic polynomial of an endomorphism is by definition that of any representing matrix (The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero), and for $A \in M_{n} (F)$ the polynomial $χ_{A} (x)$ is monic of degree $n$ ( $χ_{A} (x)$ is monic of degree $n$ ; for $n \geq 1$ its $x^{n - 1}$ coefficient is $- tr (A)$ and its constant coefficient is $(- 1)^{n} \det (A)$ , while $χ_{0 \times 0} = 1$ ). So $χ_{T}$ is monic of degree $n = \dim V$ .

[L3]

The roots in $F$ of $χ_{T}$ are exactly the eigenvalues of $T$ (For every finite-dimensional space, $σ_{F} (T)$ is exactly the set of roots in $F$ of $χ_{T}$ ).

Proof

technique · direct

1.1

Since $V \neq {0}$ , its finite dimension $n$ is positive, so [L2] makes $χ_{T}$ a nonconstant polynomial.

L2given

2.1

By [L1], $χ_{T}$ has a root $λ \in F$ , and [L3] makes $λ$ an eigenvalue of $T$ .

step 1.1L1L3

3.1

Hence every such endomorphism has an eigenvalue; the excluded zero-dimensional case would have characteristic polynomial $1$ and no eigenvalue.

step 2.1L2L3∎

Depends on

Used by

Schur’s lemma for irreducible Lie-algebra representations Corollary
Convolution of matrix coefficients on a compact group Example
Nonzero finite dimensional complex invariant subspaces have unitary eigenvectors Lemma
Finite-dimensional representations of sl₂ Theorem
Lie's theorem Theorem
Schur orthogonality Theorem

Dependency tree · two levels

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Sources

M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)