Eigenvalue Basics

Overview

Eigenvalues and eigenvectors are fundamental concepts that reveal intrinsic properties of linear transformations. They describe directions that are preserved (only scaled) by the transformation.

Definition

For a square matrix $A$, a scalar $\lambda$ is an eigenvalue and a non-zero vector $\mathbf{v}$ is an eigenvector if:

$A\mathbf{v} = \lambda\mathbf{v}$

When $A$ transforms $\mathbf{v}$, the result is just $\mathbf{v}$ scaled by $\lambda$.

Intuition

Geometric Meaning

• An eigenvector points in a direction that $A$ preserves (or reverses)
• The eigenvalue is the scaling factor along that direction
• $\lambda > 1$: stretching
• $0 < \lambda < 1$: compression
• $\lambda < 0$: reversal and scaling
• $\lambda = 0$: collapsed to zero

Example

$A = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

$A\mathbf{v} = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \end{bmatrix} = 3\begin{bmatrix} 1 \\ 0 \end{bmatrix} = 3\mathbf{v}$

So $\lambda = 3$ is an eigenvalue with eigenvector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$.

Finding Eigenvalues

From $A\mathbf{v} = \lambda\mathbf{v}$:

$A\mathbf{v} - \lambda\mathbf{v} = \mathbf{0}$

$(A - \lambda I)\mathbf{v} = \mathbf{0}$

For non-zero $\mathbf{v}$, $(A - \lambda I)$ must be singular:

$\det(A - \lambda I) = 0$

This is the characteristic equation.

Characteristic Polynomial

$p(\lambda) = \det(A - \lambda I)$

This is a polynomial of degree $n$ in $\lambda$. Its roots are the eigenvalues.

Example: Finding Eigenvalues

$A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}$

$A - \lambda I = \begin{bmatrix} 4-\lambda & 2 \\ 1 & 3-\lambda \end{bmatrix}$

$\det(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 \times 1 = 12 - 7\lambda + \lambda^2 - 2 = \lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2)$

Eigenvalues: $\lambda_1 = 5$, $\lambda_2 = 2$

Special Eigenvalues

Identity Matrix

$I$: all eigenvalues $= 1$

Diagonal Matrix

$D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$

Eigenvalues: $d_1, d_2, d_3$ (diagonal entries)

Triangular Matrix

Eigenvalues are the diagonal entries.

Properties

Property	Formula
Sum of eigenvalues	$= \text{trace}(A) = a_{11} + a_{22} + \cdots + a_{nn}$
Product of eigenvalues	$= \det(A)$
Number of eigenvalues	$\leq n$ (counting multiplicity)

Multiplicity

Algebraic Multiplicity

The number of times $\lambda$ appears as a root of the characteristic polynomial.

Geometric Multiplicity

The dimension of the eigenspace for $\lambda$:

$\dim(\text{Null}(A - \lambda I))$

Always: geometric $\leq$ algebraic multiplicity.

Eigenspace

For eigenvalue $\lambda$, the eigenspace is:

$E_\lambda = \text{Null}(A - \lambda I) = \{\mathbf{v} : A\mathbf{v} = \lambda\mathbf{v}\}$

This is a subspace containing:

• All eigenvectors for $\lambda$
• The zero vector

Complex Eigenvalues

Real matrices can have complex eigenvalues. They always appear in conjugate pairs:

• If $\lambda = a + bi$ is an eigenvalue, so is $\bar{\lambda} = a - bi$

Example: Complex Eigenvalues

$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

$\det(A - \lambda I) = \lambda^2 + 1 = 0$

$\lambda = \pm i$

This is a 90° rotation matrix.