Outline Algebraic and Geometric Multiplicity Generalized Eigenvectors Jordan Chain Jordan Normal Form Eigenvalues and Eigenvectors of Hermitian Matrix

Algebraic and Geometric Multiplicity Algebraic Multiplicity Eigenspace Associated with Eigenvalues Geometric Multiplicity

Algebraic and Geometric Multiplicity Algebraic Multiplicity ≥ Geometric Multiplicity An Example of “>”

Algebraic and Geometric Multiplicity Algebraic Multiplicity ≥ Geometric Multiplicity Proof: Ker(A - λiI) = span{u1, u2, …, nγ} A basis U = [u1, u2, …, nγ, …, uN]

Generalized Eigenvectors Eigenvalue λ of Matrix A Eigenvector of A for eigenvalue λ: Generalized eigenvector of A for eigenvalue λ for certain m: generalized eigenvector of rank m Algebraic Multiplicity: number of linearly independent Generalized eigenvectors Geometric Multiplicity: number of linearly independent eigenvectors

Jordan Chain Generalized Eigenvectors of Rank m Rank m, (A – λI)mu = 0; Jordan chain: u, (A – λI)u, (A – λI)2u, …, (A – λI)m-1u Matrix form of Jordan chain??

Jordan Normal Form Matrix A Formal proof: mathematical induction Jordan matrix: J = M-1AM J = diag(J1, J2, …, Jd), Jordan matrix Jk Formal proof: mathematical induction

Eigenvalues for Matrix Matrix A can be diagnonized All eigenvectors of A are linear independent Geometric multiplicity = algebraic multiplicity

Eigenvalues for Hermitian Matrix Hermitian Matrix A All eigenvalues are real; (A – λI) (A – λI)u = 0  uH(A – λI)(A – λI)u = 0  (A – λI)u = 0  no eigenvectors of rank ≥ 2  Algrabric multiplicity = Geometric multiplicity Hermitian Matrix A: EVD A = UHΣU, diagonal matrix Σ