# Eigenspaces and the Fundamental Theorem of Algebra

Norbert A'Campo

The following well-known theorem will be proved first, and then the so-called Fundamental Theorem of Algebra will be deduced from it.

Theorem 1   Let be a -vectorspace of finite dimension, , and let be a linear transformation. Then there exists a linear subspace in with and .

Preliminaries: 1. The angular variation is defined for , with by the equation

For with , and the angular variations obey the following additive rule:

2. For a continuous curve the angular variation along is defined to be

where , is chosen in such a way that for all with the inequality holds. The angular variation does not depend on the actual choice of since for two such choices and it follows from the additive rule: The angular variation satisfies:

3. For a closed continuous curve it follows To see this, observe:

4. For a constant curve we have . To see this, compute with .

5. For the curve , we have . Compute with .

6. For a continuous family of closed, continuous curves we have

Proof:The function is uniformly continuous. Choose in such a way that for all with the inequality holds. The function

is continuous, with values in by 3, and therefore constant. Hence

Proof of the theorem:Choose a basis , , for and consider the function

where is the linear transformation given by

We have for . For any with and the subspace satisfies: Therefore it suffices to prove the existence of a vector with and

Fig. 1.

Choose in two linear independent vectors and , and consider the family of curves given by:

We claim: there exists such that , and the vector satisfies and . We prove the claim by contradiction. Assume for all . Observe

We obtain: using the preliminaries 5, 6 and 4, which contradicts the hypothesis .

Corollary 1   Let be a vectorspace of finite dimension, , and let a linear transformation. Then there exists a linear subspace in with and . Moreover, every , , is an eigenvector, i.e. for some the equation holds. The matrix of is uppertriangular with respect to some basis of

Proof:Consider a chain of maximal length of linear subspaces in V, with and Then and induces on each quotient space a linear transformation, say such that there is no subspace in with and So, by the theorem we have Then and the subspace has the property of the corollary. A system such that is a basis of for which the matrix of is uppertriangular.

Corollary 2   (Fundamental Theorem of Algebra'') Let be a polynomial of degree . Then their exists with .

Proof:Let be a polynomial of degree . Consider the linear transformation acting on the standard basis of as follows:

Clearly and , . We conclude that . Thus for an eigenvector , with eigenvalue , we get , and hence .

Remark. The Fundamental Theorem of Algebra was proved by C. Gauss [G]. As is done in the textbooks on Linear Algebra, one can use it also to proof directly the existence of eigenvectors for endomorphism of finite dimensional vector spaces over . A first appearance of eigenvectors for symetric matrices is in the work of Cauchy on inertials elipsoids (see the textbook of P.M. Cohn [C]). The Jordan Normal Form Theorem is formulated and proved by C. Jordan [J] for general fields . If any polynomial of positiv degree factors in in polynomials of degree one, the Normal Form Theorem proofs the existence of invariant flags for endomorphisms of finite dimensional vector spaces over the field .

[C] P.M. Cohn, Algebra, Wiley, New York, 1982.

[J] Camille Jordan, Traité des substitutions et équations algébriques, Gauthiers-Villars, Paris, 1957.

[G] C. Gauss, .