Eigenvector

Let $T$ be a linear transformation or a matrix, an eigenvector for $T$ is a non-0 vector that doesn't change direction when $T$ is applied.

$\vec{v} \neq \vec{0}$ is an eigenvector for $T$ if $\exist \lambda \in \R$ s.t. $T\vec{v}=\lambda\vec{v}$

$\lambda$ is the eigenvalue corresponding to $\vec{v}$

Characteristic Polynomial

The characteristic polynomial of matrix $M$ is:

$\text{char}(A)=\det(A-\lambda I)$

• Example: Find Characteristic Polynomials

Characteristic Poly Properties

For a $n\times n$ matrix $M$:

• $\text{char}(M)$ is a $n$-degree polynomial
• Coefficient of the $\lambda^n$ term is $\pm1$ (1 if $n$ is even, -1 if odd)
• $\text{char}(M)$ evaluated at $\lambda=0$ is $\det(M)$
• The roots of $\text{char}(M)$ are precisely the eigenvalues of $M$

Eigenspace & Multiplicity

The Eigenspace of $n \times n$ matrix $A$ corresponding to $\lambda_i$ is $\text{null}(A-\lambda_i I)$

The geometric multiplicity of $\lambda_i$ is $\dim(\text{null}(A-\lambda_i I))$

The algebraic multiplicity of $\lambda_i$ is the number of times $\lambda_i$ occurs as a root of $\text{char}(A)$

$\text{geo\_mult}(\lambda)\le\text{alg\_mult}(\lambda)$

$A$ is diagonalizable $\Leftrightarrow$ sum of geometric multiplicities is $n$