Discover the intrinsic structure of matrices through eigenanalysis.
Eigenvalues & Eigenvectors
Av = λv
A is a square matrix, v ≠ 0 is the eigenvector, λ is the eigenvalue
An eigenvector of A is a nonzero vector whose direction is preserved (or reversed) under the transformation A — only its length scales by the factor λ. This captures the "natural axes" of the transformation.
The Characteristic Equation
det(A − λI) = 0
This yields a polynomial of degree n in λ
For λ = 3: (A − 3I)v = 0 → v₁ = (1, 0). For λ = 2: v₂ = (−1, 1).
Diagonalization
A = PDP⁻¹
P = [v₁ | v₂ | … | vₙ] (eigenvectors as columns)
D = diag(λ₁, λ₂, …, λₙ)
A matrix is diagonalizable if it has n linearly independent eigenvectors. Diagonal matrices are easy to work with: Aᵏ = PDᵏP⁻¹ where Dᵏ just raises each diagonal entry to the kth power. This makes computing matrix powers and exponential functions of matrices efficient.
Google PageRank: The dominant eigenvector of a link matrix ranks web pages
Vibration analysis: Eigenvalues give natural frequencies — connects to wave applications
When a matrix isn't diagonalizable, the Jordan Normal Form provides the closest alternative. For real-world computation, the Singular Value Decomposition (A = UΣVᵀ) is the most powerful factorization — it always exists and handles rectangular matrices too.