Eigenvalue Formulas & Reference

Linear Algebra for Machine Learning · Axiom Academy

Eigenvalue & Eigenvector Formulas Quick reference for all key formulas Eigenvalue-Eigenvector Equation The fundamental relationship defining eigenvalues and eigenvectors: The equation whose roots are the eigenvalues: Characteristic Polynomial (2×2 example) Expanding the determinant for a 2×2 matrix: Relationships Between Eigenvalues Product of Eigenvalues = Determinant When a matrix can be diagonalized: Computing Powers via Diagonalization Spectral Decomposition (Symmetric Matrices) Spectral Theorem & Positive Definite Matrices Spectral Theorem (Symmetric Matrices) A symmetric matrix A is positive definite if and only if: A symmetric matrix A is positive definite iff: For symmetric matrix A with eigendecomposition: An approximation to the eigenvalue:

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