# EIGENVALUES AND EIGENVECTORS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called a quadratic form in x and and we have the identit where X = form ax + hx + b = x a h h b x and A = a h h b x = X t AX, A is called the matrix of the quadratic We now rotate the x, axes anticlockwise through θ radians to new x 1, 1 axes The equations describing the rotation of axes are derived as follows: Let P have coordinates (x, ) relative to the x, axes and coordinates (x 1, 1 ) relative to the x 1, 1 axes Then referring to Figure 61: 115

2 116 CHAPTER 6 EIGENVALUES AND EIGENVECTORS 1 P x 1 R α θ O Q x Figure 61: Rotating the axes x = OQ = OP cos (θ + α) = OP(cos θ cos α sinθ sinα) = (OP cos α)cos θ (OP sinα)sin θ = OR cos θ PR sinθ = x 1 cos θ 1 sin θ Similarl = x 1 sinθ + 1 cos θ We can combine these transformation equations into the single matrix equation: x cos θ sinθ x1 =, sinθ cos θ 1 x x1 cos θ sinθ or X = PY, where X =, Y = and P = 1 sinθ cos θ We note that the columns of P give the directions of the positive x 1 and 1 axes Also P is an orthogonal matrix we have PP t = I and so P 1 = P t The matrix P has the special propert that detp = 1 cos θ sinθ A matrix of the tpe P = is called a rotation matrix sinθ cos θ We shall show soon that an real orthogonal matrix with determinant

3 61 MOTIVATION 117 equal to 1 is a rotation matrix We can also solve for the new coordinates in terms of the old ones: x1 = Y = P t cos θ sinθ x X =, sinθ cos θ 1 so x 1 = xcos θ + sinθ and 1 = xsinθ + cos θ Then X t AX = (PY ) t A(PY ) = Y t (P t AP)Y Now suppose, as we later show, that it is possible to choose an angle θ so that P t AP is a diagonal matrix, sa diag(λ 1, λ ) Then X t AX = λ x x1 1 = λ 0 λ 1 x 1 + λ 1 (61) and relative to the new axes, the equation ax + hx + b = c becomes λ 1 x 1 + λ 1 = c, which is quite eas to sketch This curve is smmetrical about the x 1 and 1 axes, with P 1 and P, the respective columns of P, giving the directions of the axes of smmetr Also it can be verified that P 1 and P satisf the equations 1 AP 1 = λ 1 P 1 and AP = λ P These equations force a restriction on λ 1 and λ For if P 1 = first equation becomes a h u1 u1 = λ h b 1 v 1 v 1 or a λ1 h h b λ 1 u1 v 1 = u1 v 1 0 0, the Hence we are dealing with a homogeneous sstem of two linear equations in two unknowns, having a non trivial solution (u 1, v 1 ) Hence a λ 1 h h b λ 1 = 0 Similarl, λ satisfies the same equation In expanded form, λ 1 and λ satisf λ (a + b)λ + ab h = 0 This equation has real roots λ = a + b ± (a + b) 4(ab h ) = a + b ± (a b) + 4h (6) (The roots are distinct if a b or h 0 The case a = b and h = 0 needs no investigation, as it gives an equation of a circle) The equation λ (a+b)λ+ab h = 0 is called the eigenvalue equation of the matrix A

4 118 CHAPTER 6 EIGENVALUES AND EIGENVECTORS 6 Definitions and examples DEFINITION 61 (Eigenvalue, eigenvector) Let A be a complex square matrix Then if λ is a complex number and X a non zero complex column vector satisfing AX = λx, we call X an eigenvector of A, while λ is called an eigenvalue of A We also sa that X is an eigenvector corresponding to the eigenvalue λ So in the above example P 1 and P are eigenvectors corresponding to λ 1 and λ, respectivel We shall give an algorithm which starts from the a h eigenvalues of A = and constructs a rotation matrix P such that h b P t AP is diagonal As noted above, if λ is an eigenvalue of an n n matrix A, with corresponding eigenvector X, then (A λi n )X = 0, with X 0, so det(a λi n ) = 0 and there are at most n distinct eigenvalues of A Conversel if det (A λi n ) = 0, then (A λi n )X = 0 has a non trivial solution X and so λ is an eigenvalue of A with X a corresponding eigenvector DEFINITION 6 (Characteristic polnomial, equation) The polnomial det (A λi n ) is called the characteristic polnomial of A and is often denoted b ch A (λ) The equation det(a λi n ) = 0 is called the characteristic equation of A Hence the eigenvalues of A are the roots of the characteristic polnomial of A a b For a matrix A =, it is easil verified that the characteristic polnomial is λ (tracea)λ+det A, where trace A = a+d is the sum c d of the diagonal elements of A 1 EXAMPLE 61 Find the eigenvalues of A = and find all eigenvectors 1 Solution The characteristic equation of A is λ 4λ + 3 = 0, or (λ 1)(λ 3) = 0 Hence λ = 1 or 3 The eigenvector equation (A λi n )X = 0 reduces to λ 1 1 λ x = 0 0,

5 6 DEFINITIONS AND EXAMPLES 119 or Taking λ = 1 gives ( λ)x + = 0 x + ( λ) = 0 x + = 0 x + = 0, which has solution x =, arbitrar Consequentl the eigenvectors corresponding to λ = 1 are the vectors, with 0 Taking λ = 3 gives x + = 0 x = 0, which has solution x =, arbitrar Consequentl the eigenvectors corresponding to λ = 3 are the vectors, with 0 Our next result has wide applicabilit: THEOREM 61 Let A be a matrix having distinct eigenvalues λ 1 and λ and corresponding eigenvectors X 1 and X Let P be the matrix whose columns are X 1 and X, respectivel Then P is non singular and P 1 λ1 0 AP = 0 λ Proof Suppose AX 1 = λ 1 X 1 and AX = λ X We show that the sstem of homogeneous equations xx 1 + X = 0 has onl the trivial solution Then b theorem 510 the matrix P = X 1 X is non singular So assume xx 1 + X = 0 (63) Then A(xX 1 + X ) = A0 = 0, so x(ax 1 ) + (AX ) = 0 Hence xλ 1 X 1 + λ X = 0 (64)

6 10 CHAPTER 6 EIGENVALUES AND EIGENVECTORS Multipling equation 63 b λ 1 and subtracting from equation 64 gives (λ λ 1 )X = 0 Hence = 0, as (λ λ 1 ) 0 and X 0 Then from equation 63, xx 1 = 0 and hence x = 0 Then the equations AX 1 = λ 1 X 1 and AX = λ X give AP = AX 1 X = AX 1 AX = λ 1 X 1 λ X λ1 0 λ1 0 = X 1 X = P 0 λ 0 λ so P 1 λ1 0 AP = 0 λ 1 EXAMPLE 6 Let A = be the matrix of example 61 Then X 1 = and X 1 = are eigenvectors corresponding to eigenvalues and 3, respectivel Hence if P =, we have 1 1 P AP = 0 3 There are two immediate applications of theorem 61 The first is to the calculation of A n : If P 1 AP = diag (λ 1, λ ), then A = Pdiag (λ 1, λ )P 1 and ( ) n n A n λ1 0 = P P 1 λ1 0 = P P 1 λ n = P λ 0 λ 0 λ n P 1 The second application is to solving a sstem of linear differential equations = ax + b d = cx + d, a b where A = is a matrix of real or complex numbers and x and c d are functions of t The sstem can be written in matrix form as Ẋ = AX, where x ẋ X = and Ẋ = = ẏ d,

7 6 DEFINITIONS AND EXAMPLES 11 We make the substitution X = PY, where Y = x1 are also functions of t and Ẋ = PẎ = AX = A(PY ), so Ẏ = (P 1 λ1 0 AP)Y = 0 λ 1 Then x 1 and 1 Y Hence x 1 = λ 1 x 1 and 1 = λ 1 These differential equations are well known to have the solutions x 1 = x 1 (0)e λ1t and 1 = 1 (0)e λt, where x 1 (0) is the value of x 1 when t = 0 If = kx, where k is a constant, then d ( ) e kt x = ke kt x + e kt = ke kt x + e kt kx = 0 Hence e kt x is constant, so e kt x = e k0 x(0) = x(0) Hence x = x(0)e kt x1 (0) x(0) However = P 1 (0) 1, so this determines x (0) 1 (0) and 1 (0) in terms of x(0) and (0) Hence ultimatel x and are determined as explicit functions of t, using the equation X = PY 3 EXAMPLE 63 Let A = Use the eigenvalue method to 4 5 derive an explicit formula for A n and also solve the sstem of differential equations d given x = 7 and = 13 when t = 0 = x 3 = 4x 5, Solution The characteristic polnomial of A is λ +3λ+ which has distinct 1 roots λ 1 = 1 and λ = We find corresponding eigenvectors X 1 = and X = Hence if P =, we have P AP = diag ( 1, ) Hence A n = ( Pdiag ( 1, )P 1) n = Pdiag (( 1) n, ( ) n )P ( 1) n = ( ) n 1 1

8 1 CHAPTER 6 EIGENVALUES AND EIGENVECTORS = ( 1) n = ( 1) n 1 3 n 1 4 n n = ( 1) n 4 3 n n 4 4 n n To solve the differential equation sstem, make the substitution X = PY Then x = x , = x The sstem then becomes ẋ 1 = x 1 ẏ 1 = 1, so x 1 = x 1 (0)e t, 1 = 1 (0)e t Now x1 (0) 1 (0) = P 1 x(0) (0) = = 11 6 so x 1 = 11e t and 1 = 6e t Hence x = 11e t + 3(6e t ) = 11e t + 18e t, = 11e t + 4(6e t ) = 11e t + 4e t For a more complicated example we solve a sstem of inhomogeneous recurrence relations EXAMPLE 64 Solve the sstem of recurrence relations given that x 0 = 0 and 0 = 1 x n+1 = x n n 1 n+1 = x n + n +, Solution The sstem can be written in matrix form as where A = X n+1 = AX n + B, 1 1 It is then an eas induction to prove that and B = 1 X n = A n X 0 + (A n A + I )B (65),

9 6 DEFINITIONS AND EXAMPLES 13 Also it is eas to verif b the eigenvalue method that A n = n 1 3 n 1 3 n n = 1 U + 3n V, where U = and V = Hence A n A + I = n U + (3n ) V = n U + (3n 1) V 4 Then equation 65 gives ( ) 1 X n = U + 3n V which simplifies to xn n = 0 1 ( n + U + (3n 1) V 4 (n n )/4 (n n )/4 Hence x n = (n n )/4 and n = (n n )/4 ) 1 REMARK 61 If (A I ) 1 existed (that is, if det(a I ) 0, or equivalentl, if 1 is not an eigenvalue of A), then we could have used the formula A n A + I = (A n I )(A I ) 1 (66) However the eigenvalues of A are 1 and 3 in the above problem, so formula 66 cannot be used there Our discussion of eigenvalues and eigenvectors has been limited to matrices The discussion is more complicated for matrices of size greater than two and is best left to a second course in linear algebra Nevertheless the following result is a useful generalization of theorem 61 The reader is referred to 8, page 350 for a proof THEOREM 6 Let A be an n n matrix having distinct eigenvalues λ 1,,λ n and corresponding eigenvectors X 1,,X n Let P be the matrix whose columns are respectivel X 1,,X n Then P is non singular and P 1 AP = λ λ λ n,

10 14 CHAPTER 6 EIGENVALUES AND EIGENVECTORS Another useful result which covers the case where there are multiple eigenvalues is the following (The reader is referred to 8, pages for a proof): THEOREM 63 Suppose the characteristic polnomial of A has the factorization det(λi n A) = (λ c 1 ) n1 (λ c t ) nt, where c 1,,c t are the distinct eigenvalues of A Suppose that for i = 1,,t, we have nullit (c i I n A) = n i For each such i, choose a basis X i1,,x ini for the eigenspace N(c i I n A) Then the matrix P = X 11 X 1n1 X t1 X tnt is non singular and P 1 AP is the following diagonal matrix P 1 AP = c 1 I n c I n c t I nt (The notation means that on the diagonal there are n 1 elements c 1, followed b n elements c,, n t elements c t ) 63 PROBLEMS Let A = Find an invertible matrix P such that P AP = diag (1, 3) and hence prove that If A = as n A n = 3n 1 A + 3 3n I, prove that A n tends to a limiting matrix /3 /3 1/3 1/3

11 63 PROBLEMS 15 3 Solve the sstem of differential equations d given x = 13 and = when t = 0 = 3x = 5x 4, Answer: x = 7e t + 6e t, = 7e t + 15e t 4 Solve the sstem of recurrence relations x n+1 = 3x n n n+1 = x n + 3 n, given that x 0 = 1 and 0 = Answer: x n = n 1 (3 n ), n = n 1 (3 + n ) a b 5 Let A = be a real or complex matrix with distinct eigenvalues c d λ 1, λ and corresponding eigenvectors X 1, X Also let P = X 1 X (a) Prove that the sstem of recurrence relations x n+1 n+1 = ax n + b n = cx n + d n has the solution xn n = αλ n 1X 1 + βλ n X, where α and β are determined b the equation α = P 1 x0 β 0 (b) Prove that the sstem of differential equations has the solution x d = ax + b = cx + d = αe λ1t X 1 + βe λt X,

12 16 CHAPTER 6 EIGENVALUES AND EIGENVECTORS where α and β are determined b the equation α = P 1 x(0) β (0) a11 a 6 Let A = 1 be a real matrix with non real eigenvalues λ = a 1 a a + ib and λ = a ib, with corresponding eigenvectors X = U + iv and X = U iv, where U and V are real vectors Also let P be the real matrix defined b P = U V Finall let a + ib = re iθ, where r > 0 and θ is real (a) Prove that AU AV = au bv = bu + av (b) Deduce that P 1 AP = a b b a (c) Prove that the sstem of recurrence relations x n+1 n+1 = a 11 x n + a 1 n = a 1 x n + a n has the solution xn = r n {(αu + βv )cos nθ + (βu αv )sinnθ}, n where α and β are determined b the equation α = P 1 x0 β 0 (d) Prove that the sstem of differential equations d = ax + b = cx + d

13 63 PROBLEMS 17 has the solution x = e at {(αu + βv )cos bt + (βu αv )sin bt}, where α and β are determined b the equation α = P 1 x(0) β (0) x x1 Hint: Let = P Also let z = x 1 + i 1 Prove that and deduce that 1 ż = (a ib)z x 1 + i 1 = e at (α + iβ)(cos bt + i sinbt) Then equate real and imaginar parts to solve for x 1, 1 and hence x, a b 7 (The case of repeated eigenvalues) Let A = and suppose c d that the characteristic polnomial of A, λ (a + d)λ + (ad bc), has a repeated root α Also assume that A αi Let B = A αi (i) Prove that (a d) + 4bc = 0 (ii) Prove that B = 0 (iii) Prove that BX 0 for some vector X ; indeed, show that X 1 0 can be taken to be or 0 1 (iv) Let X 1 = BX Prove that P = X 1 X is non singular, and deduce that AX 1 = αx 1 and AX = αx + X 1 P 1 AP = α 1 0 α 8 Use the previous result to solve sstem of the differential equations d = 4x = 4x + 8,

14 18 CHAPTER 6 EIGENVALUES AND EIGENVECTORS given that x = 1 = when t = 0 To solve the differential equation kx = f(t), k a constant, multipl throughout b e kt, thereb converting the left hand side to (e kt x) Answer: x = (1 3t)e 6t, = (1 + 6t)e 6t 9 Let A = 1/ 1/ 0 1/4 1/4 1/ 1/4 1/4 1/ (a) Verif that det(λi 3 A), the characteristic polnomial of A, is given b (λ 1)λ(λ 1 4 ) (b) Find a non singular matrix P such that P 1 AP = diag (1, 0, 1 4 ) (c) Prove that if n 1 A n = n Let A = (a) Verif that det(λi 3 A), the characteristic polnomial of A, is given b (λ 3) (λ 9) (b) Find a non singular matrix P such that P 1 AP = diag (3, 3, 9)

### Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### 3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### MATH 2030: MATRICES. Av = 3

MATH 030: MATRICES Introduction to Linear Transformations We have seen that we ma describe matrices as smbol with simple algebraic properties like matri multiplication, addition and scalar addition In

### Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

### MAT188H1S Lec0101 Burbulla

Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

### Preliminaries of linear algebra

Preliminaries of linear algebra (for the Automatic Control course) Matteo Rubagotti March 3, 2011 This note sums up the preliminary definitions and concepts of linear algebra needed for the resolution

### x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra-Lab 2

Linear Algebra-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4) 2x + 3y + 6z = 10

### 1 Quiz on Linear Equations with Answers. This is Theorem II.3.1. There are two statements of this theorem in the text.

1 Quiz on Linear Equations with Answers (a) State the defining equations for linear transformations. (i) L(u + v) = L(u) + L(v), vectors u and v. (ii) L(Av) = AL(v), vectors v and numbers A. or combine

### Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

### 4.9 Markov matrices. DEFINITION 4.3 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if. (i) a ij 0 for 1 i, j n;

49 Markov matrices DEFINITION 43 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if (i) a ij 0 for 1 i, j n; (ii) a ij = 1 for 1 i n Remark: (ii) is equivalent to AJ n

### On the general equation of the second degree

On the general equation of the second degree S Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600 113 e-mail:kesh@imscresin Abstract We give a unified treatment of the

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006

MATH 511 ADVANCED LINEAR ALGEBRA SPRING 26 Sherod Eubanks HOMEWORK 1 1.1 : 3, 5 1.2 : 4 1.3 : 4, 6, 12, 13, 16 1.4 : 1, 5, 8 Section 1.1: The Eigenvalue-Eigenvector Equation Problem 3 Let A M n (R). If

### y Area Moment Matrices 1 e y n θ n n y θ x e x The transformation equations for area moments due to a rotation of axes are:

' t Area Moment Matrices x' t 1 e n 1 t θ θ θ x e x t x n x n n The transformation equations for area moments due to a rotation of axes are: θ θ θ θ 2 2 x x ' xx cos + sin 2 x sin cos 2 2 ' ' xx sin +

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### The Inverse of a Matrix

The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

### ON THE FIBONACCI NUMBERS

ON THE FIBONACCI NUMBERS Prepared by Kei Nakamura The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### 12.3 Inverse Matrices

2.3 Inverse Matrices Two matrices A A are called inverses if AA I A A I where I denotes the identit matrix of the appropriate size. For example, the matrices A 3 7 2 5 A 5 7 2 3 If we think of the identit

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Introduction to Matrices for Engineers

Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

### Eigenvalues and Eigenvectors

Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### 0.1 Linear Transformations

.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

### Module 3F2: Systems and Control EXAMPLES PAPER 1 - STATE-SPACE MODELS

Cambridge University Engineering Dept. Third year Module 3F2: Systems and Control EXAMPLES PAPER - STATE-SPACE MODELS. A feedback arrangement for control of the angular position of an inertial load is

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### 9.3 Advanced Topics in Linear Algebra

548 93 Advanced Topics in Linear Algebra Diagonalization and Jordan s Theorem A system of differential equations x = Ax can be transformed to an uncoupled system y = diag(λ,, λ n y by a change of variables

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### System of First Order Differential Equations

CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

### Math 315: Linear Algebra Solutions to Midterm Exam I

Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Dr. Marques Sophie Linear algebra II SpringSemester 2016 Problem Set # 11

Dr. Marques Sophie Linear algebra II SpringSemester 2016 Office 519 marques@cims.nyu.edu Problem Set # 11 Justify all your answers completely (Or with a proof or with a counter example) unless mentioned

### C 1 x(t) = e ta C = e C n. 2! A2 + t3

Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

### Problems for Advanced Linear Algebra Fall 2012

Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### Chapter 7: Eigenvalues and Eigenvectors 16

Chapter 7: Eigenvalues and Eigenvectors 6 SECION G Sketching Conics B the end of this section ou will be able to recognise equations of different tpes of conics complete the square and use this to sketch

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### 5.3 Determinants and Cramer s Rule

290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### 6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### Rotation in the Space

Rotation in the Space (Com S 477/577 Notes) Yan-Bin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### 93 Analytical solution of differential equations. 1. Nonlinear differential equation. 2. Linear differential equation

1 93 Analytical solution of differential equations 1. Nonlinear differential equation 93.1. (Separable differential equation. See AMBS Ch 38 39.) Find analytical solution formulas for the following initial

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

### The Power Method for Eigenvalues and Eigenvectors

Numerical Analysis Massoud Malek The Power Method for Eigenvalues and Eigenvectors The spectrum of a square matrix A, denoted by σ(a) is the set of all eigenvalues of A. The spectral radius of A, denoted

### Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

### Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

### 7.4 Applications of Eigenvalues and Eigenvectors

37 Chapter 7 Eigenvalues and Eigenvectors 7 Applications of Eigenvalues and Eigenvectors Model population growth using an age transition matri and an age distribution vector, and find a stable age distribution

### CHARACTERISTIC ROOTS AND VECTORS

CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Harvard College. Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review

Harvard College Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu May 5, 2010 1 Contents Table of Contents 4 1 Linear Equations

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### Computational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science

Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Definition: A N N matrix A has an eigenvector x (non-zero) with

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Section 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions

Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA LINEAR ALGEBRA

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA LINEAR ALGEBRA KEITH CONRAD Our goal is to use abstract linear algebra to prove the following result, which is called the fundamental theorem of algebra. Theorem

### 1 Spherical Kinematics

ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic