Eigenvalues

Definition

The eigenvalue $\lambda$ is the factor by which an eigenvector is scaled when multiplied by a square matrix $A$. $A\mathbf{x} = \lambda \mathbf{x}$

• How to read: “The matrix A times the vector x equals the scalar lambda times the vector x.”
• Meaning: Eigenvalue $\lambda$ represents the amount of scaling (stretch, shrink, or reversal) along the invariant direction of an eigenvector.

Why It Matters

Eigenvalues quantify the forces along the “hidden axes” of stability and change—they tell you how intensely a bridge will vibrate or the magnitude of authority a web page has in a search. They allow us to simplify chaos into independent, measurable forces.

Core Concepts

• Characteristic equation (find the eigenvalues) $\det(A - \lambda I) = 0$

  • How to read: “The determinant of the quantity A minus lambda times the identity matrix equals zero.”
  • Meaning / when to use: This is a polynomial equation in λ (the characteristic polynomial). Its roots are the eigenvalues. For an n×n matrix it is degree n, so there are n eigenvalues counting multiplicity (some may be complex or repeated).

• Trace and determinant relationships (quick checks, no solving needed)

  • Sum of eigenvalues (with multiplicity) = trace(A) = sum of diagonal entries.
    • How to read: “The sum of the eigenvalues equals the trace of the matrix A.”
    • Meaning: Quick check—eigenvalues (with multiplicity) must add to the sum of diagonal entries.
  • Product of eigenvalues = det(A).
    • How to read: “The product of the eigenvalues equals the determinant of the matrix A.”
    • Meaning: Quick check—eigenvalues (with multiplicity) must multiply to $\det(A)$.

Connected Concepts

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