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Chapter 6 Eigenvalues

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**Outlines System of linear ODE (Omit) Diagonalization**

Hermitian matrices Ouadratic form Positive definite matrices

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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.

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**§6-1 Eignvalues & Eignvectors**

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**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.

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

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

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**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?

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**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.

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

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

Let

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Let , then

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

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

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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.

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§6-3 Diagonalization

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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?

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**NOT all matrices are diagonalizable**

e.g., Let If A is diagonalizable, nonsingular matrix S,

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

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**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

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**Theorem 6.3.2: Let is diagonalizable**

A has n linearly independent eigenvectors. Note : Similarity transformation Change of coordinate diagonalization

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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)

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Example: Let For Let

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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

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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

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Question: Are the following matrices diagonalizable ?

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**The Exponential of a Matrix**

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

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Note that Define

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**Suppose that A is diagonalizable with**

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

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§6-4 Hermitian Matrices

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

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Let , then e.g. ,

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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 )

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**(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,

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**(ii) Let and be two eigenpairs of A,**

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

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

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**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.

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**Proof of Schur’s Theorem**

Let be an eigenpair of A with Using the Gram-Schmidt process, construct an orthonormal basis of Let

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**Proof of Schur’s Theorem**

By the induction hypothesis (ii)

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**Theorem6.4.4: (Spectral Theorem)**

If , then unitary matrix U that diagonalizes A . Pf：By Theorem , unitary matrix , where T is upper triangular .

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**Cor.6.4.5: Let A be real symmetric matrix .**

Then (i) (ii) an orthogonal matrix U, is a diagonal matrix. proof：

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

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

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**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.

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**Def: A is said to be normal if**

Remark： Hermitian, Skew- Hermitian and Unitary matrices are all normal.

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**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.

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

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