Eigenvalues and eigenvectors

• categories: Linear algebra, Definition

Definition
Let $A\in {\mathbb{R}}^{n\times n}$ be a square matrix. A scalar $\lambda \in \mathbb{R}$ (or $\mathbb{C}$) is an eigenvalue of $A$ if there exists a non-zero vector $v\in {\mathbb{R}}^n$ (or ${\mathbb{C}}^n$), called an eigenvector, such that:
$Av=\lambda v.$
Equivalently, $\lambda$ and $v$ satisfy the equation $(A-\lambda I)v=0$, where $I$ is the identity matrix. (see also Eigendecomposition)

Intuition

• An eigenvalue $\lambda$ represents the factor by which the eigenvector $v$ is scaled under the transformation defined by $A$.
• Eigenvectors point in directions that remain unchanged (up to scaling) under the transformation $A$.

Key Properties

1. Characteristic Equation:
   The eigenvalues are the roots of the characteristic polynomial $\det (A-\lambda I)=0$.
2. Number of Eigenvalues:
   An $n\times n$ matrix has at most $n$ eigenvalues (counting multiplicities).
3. Diagonalizability:
   $A$ is diagonalizable if there exists a basis of eigenvectors, which occurs if $A$ has $n$ linearly independent eigenvectors.
4. Matrix trace and Determinant:
   • The trace of $A$, $\text{tr}(A)$, equals the sum of its eigenvalues: $\text{tr}(A)={\sum }_{i=1}^n{\lambda }_i$.
   • The determinant of $A$ equals the product of its eigenvalues: $\det (A)={\prod }_{i=1}^n{\lambda }_i$.
5. Symmetric Matrix:
   If $A$ is symmetric ($A=A^T$), all eigenvalues of $A$ are real, and its eigenvectors can be chosen to be orthogonal.
6. Singularity:
   $A$ is singular (not invertible) if and only if $0$ is an eigenvalue.