Eigenvectors

Definition

Eigenvectors are “exceptional” vectors $\mathbf{x}$ that do not change direction when multiplied by a square matrix $A$. The vector is only scaled by a factor called the eigenvalue. $A\mathbf{x} = \lambda \mathbf{x}$

• How to read: “The matrix A times the vector x equals the scalar lambda times the vector x.”
• Meaning: Eigenvector $x$ changes only in scale, not in direction, under multiplication by $A$.

Why It Matters

Complex systems look like a mess of coupled variables until you find their “eigen-directions” (eigenvectors). These are the “hidden axes” of stability and change—they tell you the independent directions in which a system naturally moves or vibrates.

Core Concepts

• Core defining equation $A\mathbf{x} = \lambda \mathbf{x}$

  • How to read: “The matrix A multiplied by the vector x equals lambda times the vector x.”
  • Meaning: x is unchanged in direction (only scaled by λ). These are the special directions of the linear transformation A.

• Finding eigenvectors for a known eigenvalue λ $(A - \lambda I)\mathbf{x} = 0 \quad \text{(solve for nonzero x in the nullspace)}$

  • How to read: “The quantity A minus lambda times I, multiplied by the vector x, equals the zero vector.”
  • Meaning: Solve the nullspace of $(A - \lambda I)$ to find all eigenvectors for that $\lambda$—an entire eigenspace (line, plane, etc.).

Connected Concepts

• Eigenvalues
• Linear Transformation
• Diagonalization
• Singular Value Decomposition
• Classification of Differential Equations
• Principal Component Analysis