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Eigenvalues & Eigenvectors

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What eigenvalues and eigenvectors are and how to compute them.

linear-algebraeigenvalueseigenvectors

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Eigenvector of a matrix A

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  1. 1Eigenvector of a matrix A
    A non-zero vector v such that Av = λv, where λ is a scalar; the vector's direction is unchanged by the transformation, only scaled
  2. 2Eigenvalue
    The scalar λ in Av = λv that represents the factor by which an eigenvector is scaled during the linear transformation
  3. 3Find eigenvalues of a matrix A
    Solve the characteristic equation det(A - λI) = 0, where I is the identity matrix
  4. 4Characteristic polynomial
    The polynomial det(A - λI); its roots are the eigenvalues of matrix A
  5. 5Find eigenvectors after finding eigenvalues
    For each eigenvalue λ, solve (A - λI)v = 0 to find the corresponding eigenvector(s) in the nullspace
  6. 6Geometric multiplicity of an eigenvalue
    The dimension of the eigenspace (nullspace of A - λI); number of linearly independent eigenvectors for that eigenvalue
  7. 7Algebraic multiplicity of an eigenvalue
    The multiplicity of λ as a root of the characteristic polynomial; how many times it appears
  8. 8Relationship between algebraic and geometric multiplicities
    Geometric multiplicity ≤ Algebraic multiplicity; equality holds if and only if the matrix is diagonalizable for that eigenvalue
  9. 9A matrix to be diagonalizable
    Can be written as A = PDP⁻¹ where D is diagonal (eigenvalues) and P contains eigenvectors as columns
  10. 10Trace of a matrix and how does it relate to eigenvalues
    Sum of diagonal elements; equals the sum of all eigenvalues (including multiplicities): tr(A) = Σλᵢ
  11. 11Determinant's relationship to eigenvalues
    det(A) = product of all eigenvalues (including multiplicities): det(A) = Πλᵢ
  12. 12Eigenvalues of a triangular matrix
    The diagonal entries of the matrix
  13. 13Eigenvalues of the identity matrix
    All eigenvalues are 1; every non-zero vector is an eigenvector
  14. 14Eigenvalues of the zero matrix
    All eigenvalues are 0; every non-zero vector is an eigenvector
  15. 15If A has eigenvalue λ, what are eigenvalues of A²?
    λ² (with same eigenvectors); generally, eigenvalues of Aⁿ are λⁿ
  16. 16If A has eigenvalue λ, what are eigenvalues of A⁻¹ (if invertible)?
    1/λ (with same eigenvectors); A is invertible iff all eigenvalues are non-zero
  17. 17Eigenvalues of a symmetric matrix
    Always real numbers; eigenvectors corresponding to distinct eigenvalues are orthogonal
  18. 18Spectral Theorem for symmetric matrices
    Every real symmetric matrix is orthogonally diagonalizable: A = QΛQ^T where Q is orthogonal, Λ is diagonal with eigenvalues
  19. 19Eigenvalues of an orthogonal matrix
    All have magnitude 1 (|λ| = 1); can be complex even if matrix is real
  20. 20Eigenspace
    The set of all eigenvectors corresponding to a particular eigenvalue λ, plus the zero vector; forms a subspace
  21. 21Power method for finding eigenvalues
    Iterative algorithm to find the dominant eigenvalue (largest magnitude) and corresponding eigenvector by repeatedly multiplying by A
  22. 22Defective matrix
    A matrix that is not diagonalizable; has fewer than n linearly independent eigenvectors (geometric multiplicity < algebraic multiplicity)
  23. 23Applications of eigenvalues in physics
    Quantum mechanics (energy levels), vibration analysis (natural frequencies), stability of differential equations, principal components
  24. 24How are eigenvalues used in Google's PageRank?
    Web pages are ranked by the dominant eigenvector of the link matrix; measures importance based on connections
  25. 25Principal Component Analysis (PCA)
    Statistical technique using eigenvectors of covariance matrix to reduce dimensionality; principal components are directions of maximum variance
  26. 26Eigenvalues in differential equations
    Determine stability and behavior of solutions; system dy/dt = Ay has solutions involving e^(λt), where λ are eigenvalues
  27. 27Gershgorin's Circle Theorem
    Every eigenvalue lies within at least one Gershgorin disc: center at aᵢᵢ, radius Σ|aᵢⱼ| (j≠i); helps bound eigenvalue locations
  28. 28Similar matrices and how do their eigenvalues relate
    Matrices A and B are similar if B = P⁻¹AP for some invertible P; similar matrices have identical eigenvalues
  29. 29Why can't eigenvectors be the zero vector?
    By definition, eigenvectors must be non-zero; Av = λv would always be satisfied by v=0 for any λ, making it meaningless

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