Eigentheory#

Eigenvalues and Eigenvectors#

Definition: Eigenvector

Let \(V\) be a vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

We say \(\mathbf{v} \in V \setminus \{\boldsymbol{0}\}\) is an eigenvector of \(f\) if there exists some \(\lambda \in F\) such that

\[f(\mathbf{v}) = \lambda \mathbf{v}.\]

Definition: Eigenvalues

Let \(V\) be a vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

We say that \(\lambda \in F\) is an eigenvalue of \(f\) if there exists some \(\mathbf{v} \in V \setminus \{\mathbf{0}\}\) such that

\[f(\mathbf{v}) = \lambda \mathbf{v}.\]

Definition: Point Spectrum

The point spectrum of \(f\) is the set of its eigenvalues.

Theorem: Number of Distinct Eigenvalues

Let \(V\) be a vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

If \(V\) has dimension \(n \in \mathbb{N}\), then \(f\) has at most \(n\) distinct eigenvalues.

Proof

TODO

One eigenvalue can be associated with multiple eigenvectors, but every eigenvector is associated with a single eigenvalue.

Theorem: Different Eigenvalues \(\implies\) Linear Independence

Let \(V\) be a vector space over a field \(F\) let \(f: V \to V\) be an endomorphism.

Any set of eigenvectors in which each element is associated with a different eigenvalue of \(f\) is linearly independent.

Proof

TODO

Theorem: Eigenvalue Invariance under Similarity

Let \(V\) be a vector space over a field \(F\) let \(f: V \to V\) be an endomorphism.

If \(\phi: V \to V\) is an automorphism, then

\[\tilde{f} \overset{\text{def}}{=} \phi^{-1} \circ f \circ \phi\]

has the same eigenvalues as \(f\).

Proof

TODO

Theorem: Eigenspace

Let \(V\) be a vector space over a field \(F\), let \(f: V \to V\) be an endomorphism and let \(\lambda \in F\) be an eigenvalue of \(f\).

The set of all eigenvectors associated with \(\lambda\) together with the zero vector is a linear subspace of \(V\).

Definition: Eigenspace

This subspace is known as the eigenspace of \(\lambda\).

Notation

\[\operatorname{Eig}_{\lambda} \qquad E_{\lambda}\]

Definition: Geometric Multiplicity

The geometric multiplicity of \(\lambda\) is the dimension of its eigenspace.

Proof

TODO

Theorem: Eigenspace as Kernel

Let \(V\) be a vector space over a field \(F\), let \(f: V \to V\) be an endomorphism and let \(\lambda \in F\) be an eigenvalue of \(f\).

The eigenspace of \(\lambda\) is the kernel of the endomorphism \(f - \lambda \operatorname{id}\):

\[E_{\lambda} = \ker (f - \lambda \operatorname{id})\]

Proof

TODO

Theorem: Intersection of Eigenspaces

Let \(V\) be a vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

If \(\lambda_1\) and \(\lambda_2\) are distinct eigenvalues of \(f\), then the intersection of their eigenspaces \(E_{\lambda_1}\) and \(E_{\lambda_2}\) is the zero vector:

\[E_{\lambda_1} \cap E_{\lambda_2} = \{\mathbf{0}\}\]

Proof

TODO

Characteristic Polynomials#

Definition: Characteristic Polynomial

Let \(V\) be a finite-dimensional vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

The characteristic polynomial of \(f\) is the polynomial given by the determinant of the endomorphism \(f - \lambda \operatorname{id}\), where \(\lambda\) is the variable:

\[\det(f - \lambda \operatorname{id})\]

Notation

\[\chi_f (\lambda) = \det(f - \lambda \operatorname{id})\]

Theorem: Eigenvalues as Roots of the Characteristic Polynomial

Let \(V\) be a finite-dimensional vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

The eigenvalues of \(f\) are precisely the roots of its characteristic polynomial.

Definition: Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.

Proof

TODO

Theorem: Algebraic and Geometric Multiplicity

Let \(V\) be a finite-dimensional vector space over a field \(F\) and let \(f: V \to V\) be an endomorphism.

The geometric multiplicity and the algebraic multiplicity of each eigenvalue \(\lambda\) of \(f\) are related as follows:

\[1\le \operatorname{geo}(\lambda)\le \operatorname{alg}(\lambda)\]

Proof

TODO

Eigendecomposition#

Definition: Diagonalizability

Let \(V\) be a vector space.

An endomorphism \(f: V \to V\) is diagonalizable if \(V\) has a Hamel basis consisting entirely of eigenvectors of \(f\).

Theorem: Diagonalizability via Direct Sum

Let \(V\) be a finite-dimensional vector space.

An endomorphism \(f: V \to V\) is diagonalizable if and only if \(V\) is the direct sum of the eigenspaces of \(f\)'s eigenvalues.

Proof

TODO

Theorem: Diagonalizability via Dimension

Let \(V\) be a finite-dimensional vector space.

An endomorphism \(f: V \to V\) is diagonalizable if and only if the sum of the dimensions of the eigenspaces of \(f\)'s eigenvalues is equal to the dimension of \(V\).

Proof

TODO

Theorem: Self-Adjoint \(\implies\) Diagonalizability

Let \(V\) be an inner product space and let \(f: V \to V\) be an endomorphism.

If \(V\) is finite-dimensional and \(f\) is self-adjoint (\(f = f^{\ast}\)), then \(f\) is diagonalizable and \(V\) has an orthonormal basis consisting of eigenvectors of \(f\).

Proof

TODO