Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors reveal the fundamental directions and scaling factors of linear transformations. These concepts are central to understanding dynamical systems, differential equations, and data analysis.

DefinitionEigenvalue and Eigenvector

Let $T: V \to V$ be a linear operator on vector space $V$ (or let $A$ be an $n \times n$ matrix). A nonzero vector $\mathbf{v} \in V$ is an eigenvector of $T$ (or $A$ ) if there exists a scalar $\lambda$ such that: $T(\mathbf{v}) = \lambda\mathbf{v} \quad \text{or} \quad A\mathbf{v} = \lambda\mathbf{v}$

The scalar $\lambda$ is called an eigenvalue of $T$ (or $A$ ), and $\mathbf{v}$ is an eigenvector corresponding to $\lambda$ .

The set of all eigenvalues is the spectrum of $T$ (or $A$ ).

Geometrically, eigenvectors are directions that are preserved by the transformation—they're only scaled, not rotated. The eigenvalue tells us the scale factor.

ExampleComputing Eigenvectors

For $A = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}$ , find eigenvalues and eigenvectors.

The characteristic equation is $\det(A - \lambda I) = 0$ : $\det\begin{bmatrix} 3-\lambda & 1 \\ 0 & 2-\lambda \end{bmatrix} = (3-\lambda)(2-\lambda) = 0$

Eigenvalues: $\lambda_1 = 3$ , $\lambda_2 = 2$ .

For $\lambda_1 = 3$ : Solve $(A - 3I)\mathbf{v} = \mathbf{0}$ : $\begin{bmatrix} 0 & 1 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \mathbf{0} \implies v_2 = 0$

Eigenvector: $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ (and any scalar multiple).

For $\lambda_2 = 2$ : Similar calculation gives $\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ .

DefinitionEigenspace

For an eigenvalue $\lambda$ of linear operator $T$ (or matrix $A$ ), the eigenspace $E_\lambda$ is: $E_\lambda = \ker(T - \lambda I) = \{\mathbf{v} \in V : T(\mathbf{v}) = \lambda\mathbf{v}\}$

For matrices: $E_\lambda = \ker(A - \lambda I) = \{\mathbf{v} : A\mathbf{v} = \lambda\mathbf{v}\}$

The eigenspace is a subspace containing all eigenvectors for $\lambda$ plus the zero vector. Its dimension is the geometric multiplicity of $\lambda$ .

DefinitionCharacteristic Polynomial

The characteristic polynomial of an $n \times n$ matrix $A$ is: $p_A(\lambda) = \det(A - \lambda I)$

This is a polynomial of degree $n$ in $\lambda$ . The eigenvalues of $A$ are precisely the roots of $p_A(\lambda) = 0$ (the characteristic equation).

Remark

Finding eigenvalues reduces to solving a polynomial equation, which for $n \geq 5$ has no general formula. However, numerical methods efficiently compute eigenvalues for large matrices. The fundamental theorem of algebra guarantees that an $n \times n$ matrix over $\mathbb{C}$ has exactly $n$ eigenvalues (counted with multiplicity).