Chapter 6 Eigenvalues

Outlines System of linear ODE (Omit) Diagonalization Hermitian matrices Ouadratic form Positive definite matrices

Motivations To simplify a linear dynamics such that it is as simple as possible. To realize a linear system characteristics e.g., the behavior of system dynamics.

§6-1 Eignvalues & Eignvectors

Example: In a town, each year 30% married women get divorced 20% single women get married In the 1st year, 8000 married women 2000 single women. Total population remains constant , where represent the numbers of married & single women after i years, respectively.

If Question: Why does converge? Why does it converge to the same limit vector even when the initial condition is different?

Ans: Choose a basis Given an initial for some for example, Question: How does one know choosing such a basis?

Def: Let . A scalar is said to be an eigenvalue or characteristic value of A if such that The vector is said to be an eigenvector or characteristic vector belonging to . is called an eigen pair of A. Question: Given A, How to compute eigenvalues & eigenvectors?

characteristic polynomial of A, of degree n in is an eigen pair of Note that, is a polynomial, called characteristic polynomial of A, of degree n in Thus, by FTA, A has exactly n eigenvalues including multiplicities. is a eigenvector associated with eigenvalue while is eigenspace of A.

Example: To find the eigenspace of 2:( i.e., )

To find the eigenspace of 3(i.e., ) Let

Let , then

Let . If is an eigenvalue of A with eigenvector Then This means that is also an eigen-pair of A.

Let . where are eigenvalues of A. (i) Let (ii) Compare with the coefficient of , we have

Theorem 6.1.1: Let A & B be n×n matrices, if B is similar to A, then and consequently A & B have the same eigenvalues. Pf: Let for some nonsingular matrix S.

§6-3 Diagonalization

Diagonalization Goal: Given find a nonsingular matrix S, such that is a diagonal matrix. Question1: Are all matrices diagonalizable? Question2: What kinds of A are diagonalizable? Question3: How to find S if A is diagonalizable?

NOT all matrices are diagonalizable e.g., Let If A is diagonalizable, nonsingular matrix S,

To answer Q2, Suppose that A is diagonalizable. nonsingular matrix S, Let This gives a condition for diagonalizability and a way to find S.

Theorem6.3.1: If are distinct eigenvalues of an matrix A with corresponding eigenvectors , then are linearly independent. Pf: Suppose that are linearly dependent not all zero, Suppose that

Theorem 6.3.2: Let is diagonalizable A has n linearly independent eigenvectors. Note : Similarity transformation Change of coordinate diagonalization

Remarks: Let , and (i) is an eigenpair of A for (ii) The diagonalizing matrix S is not unique because Its columns can be reordered or multiplied by an nonzero scalar (iii) If A has n distinct eigenvalues , A is diagonalizable. If the eigenvalues are not distinct , then may or may not diagonalizable depending on whether A has n linearly independent eigenvectors or not. (iv)

Example: Let For Let

Def: If an matrix A has fewer than n linearly independent eigenvectors,we say that A is defective. e.g. (i) is defective (ii) is defective

Example 4: Let A & B both have the same eigenvalues Nullity (A-2I)=1 The eigenspace associated with has only one dimension. A is NOT diagonalizable However, Nullity (B-2I)=2 B is diagonalizable

Question: Are the following matrices diagonalizable ?

The Exponential of a Matrix Motivation:The general solution of is The unique solution of Question: What is and how to compute ?

Note that Define

Suppose that A is diagonalizable with

Example 6: Compute Sol: The eigenvalues of A are with eigenvectors

§6-4 Hermitian Matrices

Hermitian matrices Let , then A can be written as , where e.g. ,

Let , then e.g. , 

Def: (a) A is said to be Hermitian if (b) A is said to be skew-Hermitian if (c) A is said to be unitary if ( i.e. its column vectors form an orthonormal set in )

(ii) Let and be two eigenpairs of A, Theorem6.4.1: Let , then (i) (ii) eigenvectors belonging to distinct eigenvalues are orthogonal. Pf：(i) Let be an eigenpair of A, (ii) Let and be two eigenpairs of A,

(ii) Let and be two eigenpairs of A, Theorem6.4.1 Pf： (ii) Let and be two eigenpairs of A,

Theorem: Let and then Pf：(i) Let be an eigenpair of A,

Theorem6.4.3 (Schur’s Theorem): Let , then unitary matrix , is upper triangular. Pf： The proof is by mathematical induction on n. (i) The result is obvious if n=1; (ii) Assume the hypothesis holds for k×k matrices; (iii) let A be a (k+1)×(k+1) matrix.

Proof of Schur’s Theorem Let be an eigenpair of A with Using the Gram-Schmidt process, construct an orthonormal basis of Let

Proof of Schur’s Theorem By the induction hypothesis (ii)

Theorem6.4.4: (Spectral Theorem) If , then unitary matrix U that diagonalizes A . Pf：By Theorem 6.4.3 , unitary matrix , where T is upper triangular .

Cor.6.4.5: Let A be real symmetric matrix . Then (i) (ii) an orthogonal matrix U, is a diagonal matrix. proof：

Example 4: Find an orthogonal matrix U that diagonalizes A. Sol : (i) (ii)

Example 4: Sol : (iii) By Gram-Schmidt process,

Note：If A has orthonormal eigenbasis Question:In addition to Hermitian matrices , is there any other matrices possessing orthonormal eigenbasis? Note：If A has orthonormal eigenbasis where U is unitary &diagonal.

Def: A is said to be normal if Remark： Hermitian, Skew- Hermitian and Unitary matrices are all normal.

then ,where U is unitary & diagonal. Theorem6.4.6: A is normal A possesses an orthonormal eigenbasis Pf： If A has an orthonormal eigenbasis, then ,where U is unitary & diagonal.

proof of Theorem6.4.6: By Th.6.4.3, unitary U, Compare the diagonal elements of