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

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

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

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

5. 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

6. 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??

7. 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

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

9. 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