# 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.

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Download "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."

## Transcription

1 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 n 1, Ax,Ay = x,y. (d) for any vector x R n 1, Ax = x.. Let A be an n n unitary matrix. Then prove that (a) the rows/columns of A form an orthonormal basis of the complex vector space C n. (b) for any two vectors x,y C n 1, Ax,Ay = x,y. (c) for any vector x C n 1, Ax = x.. 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. 4. Prove the statements made in Remark??.?? about orthogonal matrices. State and prove a similar result for unitary matrices. 5. Let A be an n n upper triangular matrix. If A is also an orthogonal matrix then prove that A = I n. 6. Determine an orthonormal basis of R 4 containing the vectors (1,,1,) and (,1,,1). 7. Consider the real inner product space C[ 1,1] with f,g = 1 1 f(t)g(t)dt. Prove that the polynomials 1,x, x 1, 5 x x form an orthogonal set in C[ 1,1]. Find the corresponding functions f(x) with f(x) = Consider the real inner product space C[ π,π] with f,g = orthonormal basis for L(x,sinx,sin(x+1)). π π f(t)g(t)dt. Find an 9. Let M be a subspace of R n and dimm = m. A vector x R n is said to be orthogonal to M if x,y = 0 for every y M. (a) How many linearly independent vectors can be orthogonal to M? (b) If M = {(x 1,x,x ) R : x 1 + x + x = 0}, determine a maximal set of linearly independent vectors orthogonal to M in R. 10. Determinean orthogonalbasisofl({(1,1,0,1),( 1,1,1, 1),(0,,1,0),(1,0,0,0)}) in R 4. 1

2 11. Let R n be endowed with the standard inner product. Suppose we have a vector x t = (x 1,x,...,x n ) R n with x = 1. (a) Then prove that the set {x} can always be extended to form an orthonormal basis of R n. (b) Let this basis be {x,x,...,x n }. Suppose B = (e 1 ],e,...,e n ) is the standard basis of R n and let A = [[x] B, [x ] B,..., [x n ] B. Then prove that A is an orthogonal matrix. 1. Let v,w R n,n 1 with u = w = 1. Prove that there exists an orthogonal matrix A such that Av = w. Prove also that A can be chosen such that det(a) = 1. Example 1. Let V = R and W = {(x,y,z) R : x+y z = 0}. Then it can be easily verified that {(1,1, 1)} is a basis of W as for each (x,y,z) W, we have x+y z = 0 and hence (x,y,z),(1,1, 1) = x+y z = 0 for each (x,y,z) W. Also, usingequation (??), for everyx t = (x,y,z) R, we haveu = x+y z (1,1, 1), w = ( x y+z, x+y+z, x+y+z ) and x = w+u. Let A = and B = Then by definition, P W (x) = w = Ax and P W (x) = u = Bx. Observe that A = A,B = B, A t = A, B t = B, A B = 0, B A = 0 and A+B = I, where 0 is the zero matrix of size and I is the identity matrix of size. Also, verify that rank(a) = and rank(b) = 1.. Let W = L( (1,,1) ) R. Then using Example??.??, and Equation (??), we get W = L({(,1,0),( 1,0,1)}) = L({(,1,0),(1,, 5)}), u = ( 5x y z, x+y z, x y+5z ) and w = x+y+z (1,,1) with (x,y,z) = w +u Hence, for A = and B = we have P W (x) = w = Ax and P W (x) = u = Bx. Observe that A = A,B = B, A t = A and B t = B, A B = 0, B A = 0 and A + B = I, where 0 is the zero matrix of size and I is the identity matrix of size. Also, verify that rank(a) = 1 and rank(b) =.,

3 Example 1. Let A = diag(d 1,d,...,d n ) with d i R for 1 i n. Then p(λ) = n (λ d i ) and the eigen-pairs are (d 1,e 1 ),(d,e ),...,(d n,e n ). i=1 [ ] 1 1. Let A =. Then p(λ) = (1 λ). Hence, the characteristic equation has 0 1 roots 1,1. That is, 1 is a repeated eigenvalue. But the system (A I )x = 0 for x = (x 1,x ) t implies that x = 0. Thus, x = (x 1,0) t is a solution of (A I )x = 0. Hence using Remark??.??, (1,0) t is an eigenvector. Therefore, note that 1 is a repeated eigenvalue whereas there is only one eigenvector. [ ] 1 0. Let A =. Then p(λ) = (1 λ). Again, 1 is a repeated root of p(λ) = 0. But 0 1 in this case, the system (A I )x = 0 has a solution for every x t R. Hence, we can choose any two linearly independent vectors x t,y t from R to get (1,x) and (1,y) as the two eigen-pairs. In general, if x 1,x,...,x n R n are linearly independent vectors then (1,x 1 ), (1,x ),...,(1,x n ) are eigen-pairs of the identity matrix, I n. [ ] 1 4. Let A =. Then p(λ) = (λ )(λ +1) and its roots are, 1. Verify that 1 the eigen-pairs are (,(1,1) t ) and ( 1,(1, 1) t ). The readers are advised to prove the linear independence of the two eigenvectors. [ ] Let A =. Then p(λ) = λ λ+ and its roots are 1+i,1 i. Hence, 1 1 over R, the matrix A has no eigenvalue. Over C, the reader is required to show that the eigen-pairs are (1+i,(i,1) t ) and (1 i,(1,i) t ). Exercise 4 1. Find the eigenvalues of a triangular matrix.. [ Find eigen-pairs ] [ over C, for each ] of [ the following ] matrices: [ ] 1 1+i i 1+i cosθ sinθ cosθ sinθ,, and 1 i 1 1+i i sinθ cosθ sinθ cosθ. Let A and B be similar matrices. (a) Then prove that A and B have the same set of eigenvalues. (b) If B = PAP 1 for some invertible matrix P then prove that Px is an eigenvector of B if and only if x is an eigenvector of A. 4. Let A = (a ij ) be an n n matrix. Suppose that for all i, 1 i n,. n a ij = a. Then prove that a is an eigenvalue of A. What is the corresponding eigenvector? j=1

4 5. Prove that the matrices A and A t have the same set of eigenvalues. Construct a matrix A such that the eigenvectors of A and A t are different. 6. Let A be a matrix such that A = A (A is called an idempotent matrix). Then prove that its eigenvalues are either 0 or 1 or both. 7. Let A be a matrix such that A k = 0 (A is called a nilpotent matrix) for some positive integer k 1. Then prove that its eigenvalues are all 0. [ ] [ ] 1 i 8. Compute the eigen-pairs of the matrices and. 1 0 i 0 Exercise 5 1. Let A be a skew symmetric matrix of order n+1. Then prove that 0 is an eigenvalue of A.. Let A be a orthogonal matrix (AA t = I). If det(a) = 1, then prove that there exists a non-zero vector v R such that Av = v. Exercise 6 Find inverse of the following matrices by using the Cayley Hamilton Theorem 4 i) ii) iii) Exercise 7 1. Let A,B M n (R). Prove that (a) if λ is an eigenvalue of A then λ k is an eigenvalue of A k for all k Z +. (b) if A is invertible and λ is an eigenvalue of A then 1 λ is an eigenvalue of A 1. (c) if A is nonsingular then BA 1 and A 1 B have the same set of eigenvalues. (d) AB and BA have the same non-zero eigenvalues. In each case, what can you say about the eigenvectors?. Prove that there does not exist an A M n (C) satisfying A n 0 but A n+1 = 0. That is, if A is an n n nilpotent matrix then the order of nilpotency n.. Let A M n (R) be an invertible matrix and let x t,y t R n. Define B = xy t A 1. Then prove that (a) 0 is an eigenvalue of B of multiplicity n 1 [Hint: Use Exercise 7.1d]. (b) λ 0 = y t A 1 x is an eigenvalue of multiplicity 1. (c) 1+αλ 0 is an eigenvalue of I +αb of multiplicity 1 for any α R. (d) 1 is an eigenvalue of I +αb of multiplicity n 1 for any α R. 4

5 (e) det(a+αxy t ) equals (1 + αλ 0 )det(a) for any α R. This result is known as the Shermon-Morrison formula for determinant. 4. Let A,B M (R) such that det(a) = det(b) and tr(a) = tr(b). (a) Do A and B have the same set of eigenvalues? (b) Give examples to show that the matrices A and B need not be similar. 5. Let A,B M n (R). Also, let (λ 1,u) be an eigen-pair for A and (λ,v) be an eigen-pair for B. (a) If u = αv for some α R then (λ 1 +λ,u) is an eigen-pair for A+B. (b) Give an example to show that if u and v are linearly independent then λ 1 +λ need not be an eigenvalue of A+B. 6. Let A M n (R)be aninvertible matrixwith eigen-pairs(λ 1,u 1 ),(λ,u ),...,(λ n,u n ). Thenprove thatb = {u 1,u,...,u n } formsabasisof R n (R). If[b] B = (c 1,c,...,c n ) t then the system Ax = b has the unique solution x = c 1 u 1 + c u + + c n u n. λ 1 λ λ n [ ] cosθ sinθ Exercise 8 1. Are the matrices A = and B = sinθ cosθ some θ,0 θ π, diagonalizable? [ ] cosθ sinθ sinθ cosθ for. Find the eigen-pairs of A = [a ij ] n n, where a ij = a if i = j and b, otherwise. [ ] A 0. Let A M n (R) and B M m (R). Suppose C =. Then prove that C is 0 B diagonalizable if and only if both A and B are diagonalizable. 4. Let T : R 5 R 5 be a linear operator with rank (T I) = and N(T) = {(x 1,x,x,x 4,x 5 ) R 5 x 1 +x 4 +x 5 = 0, x +x = 0}. (a) Determine the eigenvalues of T? (b) Find the number of linearly independent eigenvectors corresponding to each eigenvalue? (c) Is T diagonalizable? Justify your answer. 5. Let A be a non-zero square matrix such that A = 0. Prove that A cannot be diagonalized. [Hint: Use Remark??.] 5

6 6. Are the following matrices diagonalizable? i) , ii) 0 0 1, iii) and iv) 0 4 [ ] i. i 0 Exercise 9 1. Let A be a square matrix such that UAU is a diagonal matrix for some unitary matrix U. Prove that A is a normal matrix.. Let A M n (C). Then A = 1 (A + A ) + 1 (A A ), where 1 (A + A ) is the Hermitian part of A and 1 (A A ) is the skew-hermitian part of A. Recall that a similar result was given in Exercise??.??.. Every square matrix can be uniquely expressed as A = S+iT, where both S and T are Hermitian matrices. 4. Let A M n (C). Prove that A A is always skew-hermitian. 5. Does there exist a unitary matrix U such that U 1 AU = B where A = 0 and B = Exercise Let A be a skew-hermitian matrix. Then the eigenvalues of A are either zero or purely imaginary. Also, the eigenvectors corresponding to distinct eigenvalues are mutually orthogonal. [Hint: Carefully see the proof of Theorem??.]. Let A be a normal matrix with (λ,x) as an eigen-pair. Then (a) (A ) k x for k Z + is also an eigenvector corresponding to λ. (b) (λ,x) is an eigen-pair for A. [Hint: Verify A x λx = Ax λx.]. Let A be an n n unitary matrix. Then (a) the rows of A form an orthonormal basis of C n. (b) the columns of A form an orthonormal basis of C n. (c) for any two vectors x,y C n 1, Ax,Ay = x,y. (d) for any vector x C n 1, Ax = x. (e) λ = 1 for any eigenvalue λ of A. (f) the eigenvectors x,y corresponding to distinct eigenvalues λ and µ satisfy x,y = 0. That is, if (λ,x) and (µ,y) are eigen-pairs with λ µ, then x and y are mutually orthogonal. 6

7 [ ] [ ] Show that the matrices A = and B = to find a unitary matrix U such that A = U BU? are similar. Is it possible 5. Let A be a orthogonal matrix. Then prove the following: [ ] cosθ sinθ (a) if det(a) = 1, then A = for some θ, 0 θ < π. That is, A sinθ cosθ counterclockwise rotates every point in R by an angle θ. [ ] cosθ sinθ (b) if deta = 1, then A = for some θ, 0 θ < π. That is, sinθ cosθ A reflects every point in R about a line passing through origin. Determine this line. [ Or equivalently, ] there exists a non-singular matrix P such that 1 0 P 1 AP = Let A be a orthogonal matrix. Then prove the following: (a) if det(a) = 1, then A is a rotation about a fixed axis, in the sense that A has an eigen-pair (1,x) such that the restriction of A to the plane x is a two dimensional rotation in x. (b) if deta = 1, then A corresponds to a reflection through a plane P, followed by a rotation about the line through origin that is orthogonal to P Let A = 1 1. Findanon-singularmatrix P such thatp 1 AP = diag (4,1,1). 1 1 Use this to compute A Let A be a Hermitian matrix. Then prove that rank(a) equals the number of nonzero eigenvalues of A. Exercise Let A M n (R) be an invertible matrix. Prove that AA t = PDP t, where P is an orthogonal and D is a diagonal matrix with positive diagonal entries Let A = 0 1, B = and U = Prove that A and B are unitarily equivalent via the unitary matrix U. Hence, conclude that the upper triangular matrix obtained in the Schur s Lemma need not be unique.. Prove Remark??. 4. Let A be a normal matrix. If all the eigenvalues of A are 0 then prove that A = 0. What happens if all the eigenvalues of A are 1? 7

8 5. Let A be an n n matrix. Prove that if A is (a) Hermitian and xax = 0 for all x C n then A = 0. (b) a real, symmetric matrix and xax t = 0 for all x R n then A = 0. Do these results hold for arbitrary matrices? We complete this chapter by understanding the graph of ax +hxy +by +fx+gy+c = 0 for a,b,c,f,g,h R. We first look at the following example. Example 1 Sketch the graph of x +4xy +y = 5. Solution: Note that x + 4xy + y = [x, y] [ ] matrix are (5,(1,1) t ), (1,(1, 1) t ). Thus, [ ] [ = Now, let u = x+y and v = x y. Then 1 [ ] [5 ] [ ] [ ][ ] x x +4xy +y = [x, y] y [ ] [5 ] [ = [x, y] = [ u, v ][ ][ ] 5 0 u = 5u +v. 0 1 v ][ ] x and the eigen-pairs of the y 1 ] [x ] Thus, the given graph reduces to 5u +v = 5 or equivalently to u + v = 1. Therefore, the 5 given graph represents an ellipse with the principal axes u = 0 and v = 0 (correspinding to the line x+y = 0 and x y = 0, respectively). See Figure 1. y y = x y = x Figure 1: The ellipse x +4xy +y = 5. 8

9 We now consider the general conic. We obtain conditions on the eigenvalues of the associated quadratic form, defined below, to characterize conic sections in R (endowed with the standard inner product). Definition 1 (Quadratic Form) Let ax + hxy + by + gx + fy + c = 0 be the equation of a general conic. The quadratic expression ax +hxy +by = [ x, y ][ ][ ] a h x h b y is called the quadratic form associated with the given conic. Theorem 14 For fixed real numbers a,b,c,g,f and h, consider the general conic Then prove that this conic represents 1. an ellipse if ab h > 0,. a parabola if ab h = 0, and ax +hxy +by +gx+fy +c = 0.. a hyperbola if ab h < 0. [ ] a h Proof: Let A =. Then ax + hxy + by = [ x y ] [ ] x A is the associated h b y quadratic form. As A is a symmetric matrix, by Corollary??, the eigenvalues λ 1,λ of A are both real, the corresponding eigenvectors u 1,u are orthonormal and A is unitarily diagonalizable with A = [ ] [ ] [ ] [ ] [ ][ ] λ 1 0 u t u 1 u 1 u u t 0 λ u t. Let = 1 x v u t. Then y ax +hxy+by = λ 1 u +λ v and the equation of the conic section in the (u,v)-plane, reduces to λ 1 u +λ v +g 1 u+f 1 v +c = 0. (1) Now, depending on the eigenvalues λ 1,λ, we consider different cases: 1. λ 1 = 0 = λ. Substituting λ 1 = λ = 0 in Equation (1) gives the straight line g 1 u+f 1 v +c = 0 in the (u,v)-plane.. λ 1 = 0,λ > 0. As λ 1 = 0, det(a) = 0. That is, ab h = det(a) = 0. Also, in this case, Equation (1) reduces to λ (v +d 1 ) = d u+d for some d 1,d,d R. To understand this case, we need to consider the following subcases: (a) Let d = d = 0. Then v +d 1 = 0 is a pair of coincident lines. 9

10 (b) Let d = 0, d 0. d i. If d > 0, then we get a pair of parallel lines given by v = d 1 ± λ. ii. If d < 0, the solution set of the corresponding conic is an empty set. (c) Ifd 0.Then thegivenequationisoftheformy = 4aX forsometranslates X = x+α and Y = y +β and thus represents a parabola.. λ 1 > 0 and λ < 0. In this case, ab h = det(a) = λ 1 λ < 0. Let λ = α with α > 0. Then Equation (1) can be rewritten as λ 1 (u+d 1 ) α (v +d ) = d for some d 1,d,d R () whose understanding requires the following subcases: (a) Let d = 0. Then Equation () reduces to ( λ1 (u+d 1 )+ α (v+d )) ( λ1 (u+d 1 ) ) α (v +d ) = 0 or equivalently, a pair of intersecting straight lines in the (u, v)-plane. (b) Let d 0. In particular, let d > 0. Then Equation () reduces to λ 1 (u+d 1 ) d α (v +d ) d = 1 or equivalently, a hyperbola in the (u,v)-plane, with principal axes u+d 1 = 0 and v+d = λ 1,λ > 0. In this case, ab h = det(a) = λ 1 λ > 0 and Equation (1) can be rewritten as λ 1 (u+d 1 ) +λ (v +d ) = d for some d 1,d,d R. () We consider the following three subcases to understand this. (a) Let d = 0.Then Equation()reduces toapair ofperpendicular lines u+d 1 = 0 and v +d = 0 in the (u,v)-plane. (b) Let d < 0. Then the solution set of Equation () is an empty set. (c) Let d > 0. Then Equation () reduces to the ellipse λ 1 (u+d 1 ) d + α (v +d ) d = 1 whose principal axes are u+d 1 = 0 and v +d = 0. [ ] x Remark 15 Observe that the condition = [ ] [ ] u u 1 u implies that the principal y v axes of the conic are functions of the eigenvectors u 1 and u. 10

11 Exercise 16 Sketch the graph of the following surfaces: 1. x +xy +y 6x 10y =.. x +6xy +y 1x 6y = 5.. 4x 4xy +y +1x 8y = x 6xy +5y 10x+4y = 7. As a last application, we consider the following problem that helps us in understanding the quadrics. Let ax +by +cz +dxy +exz +fyz +lx+my +nz +q = 0 (4) be a general quadric. Then to get the geometrical idea of this quadric, do the following: a d e l x 1. Define A = d b f, b = m and x = y. Note that Equation (4) can be e f c n z rewritten as x t Ax+b t x+q = 0.. As A issymmetric, find an orthogonalmatrix P such that P t AP = diag(λ 1,λ,λ ).. Let y = P t x = (y 1,y,y ) t. Then Equation (4) reduces to λ 1 y 1 +λ y +λ y +l 1y 1 +l y +l y +q = 0. (5) 4. Depending on which λ i 0, rewrite Equation (5). That is, if λ 1 0 then rewrite ) ( ) λ 1 y1 +l 1y 1 as λ 1 (y 1 + l 1 λ 1 l. 1 λ 1 5. Use the condition x = Py to determine the center and the planes of symmetry of the quadric in terms of the original system. Example 17 Determine the following quadrics 1. x +y +z +xy +xz +yz +4x+y +4z + = 0.. x y +z +10 = Solution: For Part 1, observe that A = 1 1, b = and q =. Also, the orthonormal matrix P = 6 1 and P t AP = Hence, the quadric reduces to 4y1 +y +y + 10 y 1 + y 6 y + = 0. Or equivalently to 4(y ) +(y + 1 ) +(y 1 6 ) =

12 So, the standard form of the quadric is 4z1 +z +z = 9, where the center is given by 1 (x,y,z) t = P( 5 4, 1 1, 6 ) t = (, 1, )t. 0 0 For Part, observe that A = 0 1 0, b = 0 and q = 10. In this case, we can rewrite the quadric as y 10 x 10 z 10 = 1 which is the equation of a hyperboloid consisting of two sheets. The calculation of the planes of symmetry is left as an exercise to the reader. 1

### 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,

### 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

### 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

### LINEAR ALGEBRA. September 23, 2010

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

### 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

### 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

### 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

### 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

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### 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

Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### 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

### 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

### 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

### Applied Linear Algebra I Review page 1

Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

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

### APPLICATIONS. are symmetric, but. are not.

CHAPTER III APPLICATIONS Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes In symbols, A t =

### 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)(

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### EIGENVALUES AND EIGENVECTORS

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

### 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?

### 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

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### 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 ) =

### 1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.

Math Exam - Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### 6. Cholesky factorization

6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### 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

### 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

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### 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

### 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,

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### 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

### LINEAR ALGEBRA AND MATRICES M3P9

LINEAR ALGEBRA AND MATRICES M3P9 December 11, 2000 PART ONE. LINEAR TRANSFORMATIONS Let k be a field. Usually, k will be the field of complex numbers C, but it can also be any other field, e.g., the field

### 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

### 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

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### 42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections

2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### 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

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### 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

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### 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

### 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

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### R mp nq. (13.1) a m1 B a mn B

This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### 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

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### Math 6112: Fall Advanced Linear Algebra

Math 6112: Fall 2011. Advanced Linear Algebra Luca Dieci 1 December 7, 2011 1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 U.S.A. (dieci@math.gatech.edu).please bring mistakes

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### Linear Algebra Problems

Math 504 505 Linear Algebra Problems Jerry L. Kazdan Note: New problems are often added to this collection so the problem numbers change. If you want to refer others to these problems by number, it is

### INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

### 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

Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses.

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### Determinants. Chapter Properties of the Determinant

Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### 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

### Linear Algebra Review. Vectors

Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

### Lecture 6: The Group Inverse

Lecture 6: The Group Inverse The matrix index Let A C n n, k positive integer. Then R(A k+1 ) R(A k ). The index of A, denoted IndA, is the smallest integer k such that R(A k ) = R(A k+1 ), or equivalently,

### 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 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

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### 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:

### 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

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### 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

### 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

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Linear Least Squares

Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

### 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

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### 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

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### 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 +

### Linear Algebra: Determinants, Inverses, Rank

D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of