Eigentheory

Eigenvalues and Eigenvectors

Definition: Eigenvalue

Let $A\in F^{n\times n}$ be a square matrix and let $\lambda \in F$.

We say that $\lambda$ is an eigenvalue of $A$ if there exists at least one non-zero column vector $\vec{v}\in F^n$ such that

$$A\vec{v}=\lambda \vec{v}.$$

Definition: Eigenvector

Let $A\in F^{n\times n}$ be a square matrix and let $\vec{v}\in F^n$ be a non-zero column vector.

We say that $\vec{v}$ is an eigenvector of $A$ if there is some $\lambda \in F$ such that

$$A\vec{v}=\lambda \vec{v}.$$

An eigenvalue may have more than one eigenvector, but each eigenvector always belongs to a single eigenvalue.

Definition: Eigenspace

Let $A\in F^{n\times n}$ be a square matrix.

The eigenspace of an eigenvalue $\lambda$ is the union of the set of all eigenvectors belonging to $\lambda$ together with the zero vector $\vec{0}$.

$$\{\vec{v}\in F^n\mid A\vec{v}=\lambda \vec{v}\}$$

Theorem: Structure of Eigenspaces

The eigenspace of each eigenvalue is a subspace of $F^n$.

PROOF

TODO

Definition: Geometric Multiplicity

The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of its eigenspace.

Notation

$$\mathrm{geo}(\lambda )$$

Theorem: Number of Distinct Eigenvalues

An $n\times n$-matrix has at most $n$ different eigenvalues.

PROOF

TODO

Theorem: Linear Independence of Eigenvectors

Let $A\in F^{n\times n}$ be a matrix.

If ${\vec{e}}_1,\cdots ,{\vec{e}}_r$ are eigenvectors which belong to $r$ different eigenvalues ${\lambda }_1,\cdots ,{\lambda }_r$, then ${\vec{e}}_1,\cdots ,{\vec{e}}_r$ are linearly independent.

PROOF

TODO

Characteristic Polynomials

Definition: Characteristic Polynomial

The characteristic polynomial $p_A(x)$ of a square matrix $A\in F^{n\times n}$ the polynomial obtained by the expression for the following determinant, where $I_n$ is the identity matrix:

$$p_A(x)\stackrel{\text{def}}{=}\det (A-x{I}_n),$$

Theorem: Degree of the Characteristic Polynomial

The degree of the characteristic polynomial of an $n\times n$-matrix is $n$.

PROOF

TODO

Theorem: Roots of the Characteristic Polynomial

The roots of the characteristic polynomial of a square matrix $A\in F^{n\times n}$ are precisely the eigenvalues of $A$.

$$\det (A-x{I}_n)=0$$

PROOF

TODO

Definition: Algebraic Multiplicity

If the characteristic polynomial of a square matrix $A$ has a linear factorization

$$p_A(x)=(x-{\lambda }_1{)}^{k_1}\cdots (x-{\lambda }_r{)}^{k_r}$$

over the field $F$, where ${\lambda }_1,\cdots ,{\lambda }_r(r\le n)$ are the distinct eigenvalues of $A$, then we call $k_i$ the algebraic multiplicity of ${\lambda }_i$.

Notation

$$\mathrm{alg}({\lambda }_i)$$

Theorem: Algebraic and Geometric Multiplicity

The geometric multiplicity and the algebraic multiplicity of each eigenvalue $\lambda$ of a square matrix $A\in F^{n\times n}$ obey the following inequality:

$$1\le \mathrm{geo}(\lambda )\le \mathrm{alg}(\lambda )$$

PROOF

TODO

Theorem: Sum of the Eigenvalues

The distinct eigenvalues ${\lambda }_1,\cdots ,{\lambda }_r$ of a square matrix $A\in F^{n\times n}$ and their algebraic multiplicities can be used to calculate the trace of $A$ as follows:

$$\sum _{i=1}^r{\lambda }_i\cdot \mathrm{alg}({\lambda }_i)=\mathrm{Tr}(A)$$

PROOF

TODO

Theorem: Product of the Eigenvalues

The distinct eigenvalues ${\lambda }_1,\cdots ,{\lambda }_r$ of a square matrix $A\in F^{n\times n}$ and their algebraic multiplicities can be used to calculate the determinant of $A$ as follows:

$$\prod _{i=1}^r{\lambda }_i^{\mathrm{alg}({\lambda }_i)}=\det (A)$$

PROOF

TODO

Eigendecomposition

Definition: Diagonalizable Matrix

A square matrix $A\in F^{n\times n}$ is diagonalizable if it is similar to a diagonal matrix $\Lambda \in F^{n\times n}$.

Theorem: Eigendecomposition

An $n\times n$-matrix $A$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors ${\vec{e}}_1,\cdots ,{\vec{e}}_n$.

In that case, $A$ can be written as

$$A=Q\Lambda Q^{-1},$$

where the $k$-th column of $Q$ is ${\vec{e}}_k$ and $\Lambda =\mathrm{diag}({\lambda }_1,\cdots ,{\lambda }_n)$ is the diagonal matrix whose $k$-th diagonal entry is the eigenvalue ${\lambda }_k$ to which ${\vec{e}}_k$ belongs.

PROOF

TODO

Definition: Eigendecomposition

We call $Q\Lambda Q^{-1}$ the eigendecomposition of $A$.