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**CHAPTER SIX 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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**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 be the women numbers at year i, where represent married & single women respectively.

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If Question: Why does converges? Why does it converges to the same limit 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 is singular 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: Let are eigenvalues of A. To find eigenspace of 2:( i.e., )

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**To find eigenspaces of 3(i.e., )**

Let

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Let Then is an eigenvalue of A. has a nontrivial solution. is singular. loses rank.

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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 Then and consequently A & B have the same eigenvalues. Pf: Let for some nonsingular matrix S.

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Diagonalization Goal: Given find nonsingular matrix S 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 A is diagonalizable. nonsingular matrix S Let are eigenpair of A for This gives a condition for and diagonalizability and a way to find S.

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**Theorem 6.3.2: Let is diagonalizable**

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

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**Theorem6.3.1: If are distinct eigenvalues of a**

matrix A with corresponding eigenvectors , then are linear independent. Pf: Suppose are linear independent not all zero Suppose are distinct. are linear independent.

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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 or not A has n linear independent eigenvectors. (iv)

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Example: Let For Let

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

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Example: 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: Is the following matrix diagonalizable ?

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**The Exponential of a Matrix Motiration :**

Motiration: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 A is diagonalizable with**

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Example: Compute Sol: The eigenvalues A are with eigenvectors

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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 matrix A is said to be Hermitian if A is said to be skew-Hermitian if A is said to be unitary if ( → its column vectors form an orthonormal set in )

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

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 with

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Theorem: Let Then Pf：(i) Let be an eigenpair of A is pure-imaginary

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**Theorem6.4.3: (Schur`s Theorem) Let**

Then unitary matrix U is upper triangular Pf：Let be an eigenpair of A with Choose to be such that is unitary Choose Chose to be unitary Continue this process , we have the theorem

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

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

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

Then (i) (ii) an orthogonal matrix U is a diagonal matrix Remark：If A is Hermitian , then , by Th6.4.4 , Complete orthonormal eigenbasis

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Example: Find an orthogonal matrix U that diagonalizes A Sol : (i) (ii) (iii)By Gram-Schmidt Process The columns of form an orthogonormal eigenbasis (WHY?)

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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 Hermitian & D is 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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**Theorem6.4.6: A is normal A possesses**

orthonormal eigenbasis Pf： have proved By Th , unitary U is upper triangular T is also normal Compare the diagonal elements of T has orthonormal eigenbasis(WHY?)

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**Singular Value Decomposition(SVD) :**

Theorem : Let with rank(A)=r Then unitary matrices With Where

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Remark:In the SVD The scalars are called singular values of A Columns of U are called left singular vectors of A Columns of V are called right singular vectors of A

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**Pf：Note that , is Hermitian**

& Positive semidefinite with unitary matrix V where Define (1) (2) Define (3) Define is unitary

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Remark:In the SVD The singular values of A are unique while U&V are not unique Columns of U are orthonormal eigenbasis for Columns of V are orthonormal eigenbasis for is an orthonormal basis for

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**rank(A) = number of nonzero singular values**

but rank(A) ≠ number of nonzero eigenvalues for example

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Example : Find SVD of Sol : An orthonormal eigenbasis associate with can be found as Find U is orthogonal A set of candidate for are Thus

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Lemma6.5.2 : Let be orthogonal . Then Pf :

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Cor : Let be the SVD of A . Then

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**We`ll state the next result without proof :**

Theorem6.5.3 : H.(1) be the SVD of A (2) C :

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**Application : Digital Image Processing**

(especially efficient for matrix which has low rank)

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**Quadratic Forms : To classify the type of quadratic surface (line)**

Optimization : An application to the Calculus

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**Def : A quadratic equation in two variables x & y**

is an equation of the form

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**Standard forms of conic sections**

(ii) (iii) (iv) Note : Is there any difference between the eigenvalues A of the quadratic form ?

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**Goal : Try to transform the quadratic equation**

into standard form by suitable translation and rotation

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Example : (No xy term) The eigenvalues of the quadratic terms are 9 , 4 → ellipse → →

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**→ By direct computation**

Example : (Have xy term) → → By direct computation is orthogonal ( why does such U exist ? ) → Let the original equation becomes

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Example : → → Let or

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Optimization : Let It is known from Taylor’s Theorem of Calculus that Where is Hessian matrix is local extremum If then is a local minimum

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**Def : A real symmetric matrix A is said to be**

(i) Positive definite denoted by (ii) Negative definite denoted by (iii) Positive semidefinite denoted by (iv) Negative semidefinite denoted by example : is indefinite question : Given a real symmetric matrix , how to determine its definiteness efficiently ?

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Theorem6.5.1: Let Then Pf : let be eigenpair of A Suppose Let be an orthonormal eigen-basis of A (Why can assume this ? )

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**Example: Find local extrema of**

Sol : Thus f has local maximum at while are saddle points

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**Positive Definite Matrices :**

Property I : P is nonsingular Property II : and all the leading principal submatrices of A are positive definite

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Property III : P can be reduced to upper triangular form using only row operation III and the pivots elements will all be positive Sketch of the proof : & determinant is invariant under row operation of type III Continue this process , the property can be proved

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Property IV : Let Then (i) A can be decompose as A=LU where L is lower triangular & U is upper triangular (ii) A can be decompose as A=LU where L is lower triangular & U is upper triangular with all the diagonal element being equal to 1 , D is an diagonal matrix Pf : by Gaussian elimination and the fact that the product of two lower (upper) triangular matrix is lower (upper) triangular

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Example: Thus A=LU Also A=LDU with

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Property V : Let If Pf : LHS is lower triangular & RHS is upper triangular with diagonal elements 1

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Property VI : Let Then A can be factored into where D is a diagonal matrix & L is lower triangular with 1’s along the diagonal Pf : Since the LDU representation is unique

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**Property VII : (Cholesky decomposition)**

Let Then A can be factored into where L is lower triangular with positive diagonal Hint :

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**Example: We have seen that**

Note that Define we have the Choleoky decomposition

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**Theorem6.6.1 : Let , Then the followings are equivalent: (i) A>0 **

(ii) All the leading principal submatrices have positive determinants. (iii) A ～ U only using elementary row operation of type III. And the pivots are all positive , where U is an upper triangular matrix. (iv) A has Cholesky decomposition LLT. (v) A can be factored into BTB for some nonsingular matrix B row Pf : We have shown that (i) (ii) (iii) (iv) In addition , (iv) (v) (i) is trivial

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**Housholder Transformation :**

Def : Then the matrix is called Housholder transformation Geometrical lnterpretation: Q is symmetric , Q is orthogonal , What is the eigenvalues , eigenvectors and determinant of Q ?

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Given , Find QR factorization

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Theorem : Let and be a SVD of A with Then Pf : Cor : Let be nonsingular with singular values Then and

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**Application : In solving What is the effect of the**

Solution when present measurement error ?

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**is said to be the condition number of A**

If A is orthogonal then This means that , due to the error in b the deviation of the associated solution of is minimum if A is orthogonal

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Example : A is close to singular Note that , is the solution for and is the solution for What does this mean ? Similarly , i.e. small deviation in x results in large deviation in b This is the reason why we use orthogonal factorization in Numerical solving Ax=b

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