Eigenvalue

linear-algebra

Definition

Eigenvalue

Let $f:V\to V$ be a linear mapping over a field $F$. A scalar $\lambda \in F$ is an eigenvalue of $f$ if there exists a non-zero vector $\mathbf{v}\in V$ such that:

$f(\mathbf{v})=\lambda \mathbf{v}$

For a square matrix $A\in F^{n\times n}$, $\lambda$ is an eigenvalue if there exists $\mathbf{v}\ne 0$ such that $A\mathbf{v}=\lambda \mathbf{v}$. This condition is satisfied if and only if $\lambda$ is a root of the characteristic polynomial:

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

Algebraic and Geometric Properties

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

Geometric Multiplicity: The dimension of the nullspace of $(\lambda I-A)$, representing the number of linearly independent eigenvectors associated with $\lambda$. It holds that

$$1\le \text{geometric multiplicity}\le \text{algebraic multiplicity}$$

Applications: Eigenvalues are fundamental to PCA, where they quantify the variance captured by each principal direction, and to spectral clustering.