Summary

Eigenvalues and Eigenvectors

Definition

Eigenvalues and Eigenvectors

Let $A$ be a square matrix. A scalar $\lambda$ is called an eigenvalue of $A$ if there exists a nonzero vector $v$ such that $Av=\lambda v$. Such a vector $v$ is called an eigenvector corresponding to the eigenvalue $\lambda$.

Definition

characteristic polynomial of $A$ is $p(\lambda )=\text{det}\left(A-\lambda I\right)$ where $I$ is the identity matrix. The roots of this polynomial are the eigenvalues of $A$. And the eigenvectors are the solutions to the equation $\left(A-\lambda I\right)v=0$ which is called the characteristic equation.

Multiplicity of the eigenvalue

The multiplicity of an eigenvalue $\lambda$ in characteristic equation is called the algebraic multiplicity of the eigenvalue denoted by $m(\lambda )$.

Eigenspace

The set of all eigenvectors $(v_1,v_2,\dots ,v_n)$corresponding to the eigenvalue $\lambda$ is called the eigenspace of $\lambda$ denoted by $E_{\lambda }=\text{span}(v_1,v_2,\dots ,v_n)$.

Geometric Multiplicity of Eigenvalue

The dimension of the eigenspace of an eigenvalue $\lambda$ is called the geometric multiplicity of $\lambda$. It is denoted by $g(\lambda )=\text{dim}(E_{\lambda })$.

Algebraic and Geometric Multiplicity of the Eigenvalue

$m(\lambda )\ge g(\lambda )$.

Linearly independent in Eigenvectors

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Diagonalization

Diagonalizable Matrix

A square matrix $A$ is called diagonalizable if there exists an invertible matrix $P$ such that $P^{-1}AP=D$ where $D$ is a diagonal matrix. The matrix $P$ is called the matrix of eigenvectors of $A$ and $D$ is the diagonal matrix of eigenvalues of $A$.