Eigenvectors and Eigenvalues

Eigenvectors of a transformation are vectors that are only scaled by the transform. Each eigenvectors has an eigenvalues which the scaler that it is scaled by.

Definition

• T is a linear operator on $V$, $v\in V$. $v\ne 0$ is an eigenvector of T if $\exists \lambda$ scalar $T(v)=\lambda v$ ($\lambda$ is the corresponding eigenvalue).
• $A\in M_{n\times n}(F),v\in F$. $v\ne 0$ is an eigenvector if it is an eigenvector of $L_A$
  • $⟺Av=\lambda v$ for some scalar $\lambda$

Simple Properties

• Eigenvectors are unique up to a scalar multiple
  • If $v$ eigenvector corresponding to $\lambda$, then so is $tv,t\in F$
• Eigenvectors carry between changes to the coordinate system
• The set of an single corresponding eigenvector for each distinct eigenvalue is linearly independent

Diagonalizability

• Definition
  • $T$ linear operator is diagonalizable if $\exists \beta$ for $V$ such that $[T{]}_{\beta }$ is a diagonal matrix
  • $A$ square matrix is diagonalizable if $L_A$ is diagonalizable
• Properties
  • $T$ linear operator on finite dimensional vector space $V$ is diagonalizable
    • $⟺$ $\exists \beta$ ordered basis for $V$ consisting of eigenvectors of $T$
  • If $T$ is diagonalizable, $\beta =\{v_1,\dots ,v_n\}$ ordered basis of eigenvectors, $D=[T{]}_{\beta }$
    • $⟹$ $D$ is diagonal matrix and $D_{jj}$ is the eigenvalue corresponding to $v_j$
  • $T$ linear operator in n-dimensional vector space $V$
    • $T$ has $n$ distinct eigenvalues $⟹T$ diagonalizable
  • $A$ diagonalizable $⟹Q^{-}1AQ=D$ where $Q$ is the eigenvalues

Characteristic Polynomial

• $f(\lambda )=\det (A-\lambda I)$
  • For linear operator $T$, $A=[T{]}_{\beta }$
• The eigenvalues of matrix are zeros of the characteristic polynomial
• Properties for $A\in M_{n\times n}(F)$
  • Characteristic polynomial of $A$ is a polynomial of degree $n$ with leading coefficient $(-1{)}^n$
  • $A$ has at most $n$ distinct eigenvalues (because can only have at most $n$ distinct roots)
• Restatement in the context of transforms
  • T linear operator on $V$. $\lambda$ eigenvalues of T.
  • $v\in V$ is eigenvector of $T$ corresponding to $\lambda ⟺v\in N(T-\lambda I)$
• The characteristic polynomial for any diagonalizable linear operator splits (factors completely)
• The algebraic multiplicity of $\lambda$ is the largest positive integer $k$ for which $(t-\lambda {)}^k$ is a factor of $f(\lambda )$

Eigenspace

• $E_{\lambda }=\{x\in V:T(x)=\lambda x\}=N(T-\lambda I)$
  • A set of the zero vector and all eigenvectors for a given eigenvector
• Facts
  • Let $\lambda$ be an eigenvalue of $T$ having an algebraic multiplicity $m⟹1\le \dim (E_{\lambda })\le m$
  • ${\lambda }_1,\dots ,{\lambda }_k$ distinct eigenvalues of $T$. For each $i=1,\dots ,k$ let $S_i$ denote a finite linearly independent subset of $E_{{\lambda }_i}⟹S=\bigcup _{i=1}^k{S}_i$ is a linearly independent subset of $V$.
  • $T$ diagonalizable $⟺$ multiplicity of ${\lambda }_i=\dim (E_{{\lambda }_i})$
  • $T$ diagonalizable $⟹{\beta }_i$ ordered basis for $E_{{\lambda }_i}$
  • $\beta =\bigcup _{i=1}^k{\beta }_i$ is an ordered basis for $E_{{\lambda }_i}$