8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

Size: px
Start display at page:

Download "8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS"

Transcription

1 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars are real numbers. A complex vector space s one n whch the scalars are complex numbers. So, f v, v 2,..., v m are vectors n a complex vector space, then a lnear combnaton s of the form c v c 2 v 2 where the scalars c are complex numbers. The complex verson of R n, c 2,..., c s the complex vector space m consstng of ordered n-tuples of complex numbers. So, a vector n has the form It s also convenent to represent vectors n v a b a 2 b 2. a n b n. c m v m v a b, a 2 b 2,..., a n b n. by column matrces of the form As wth R n, the operatons of addton and scalar multplcaton n are performed component by component. EXAMPLE Vector Operatons n Let v 2, 3 and u 2, 4 be vectors n the complex vector space C 2. Determne each vector. (a) v u (b) 2 v (c) 3v 5 u Soluton (a) In column matrx form, the sum v u s v u (b) Because and 2 3 7, you have 2 v 2 2, 3 5, 7. (c) 3v 5 u 3 2, 3 5 2, 4 (c) 3 6, , 2 4 (c) 2,

2 494 CHAPTER 8 COMPLEX VECTOR SPACES Many of the propertes of R n are shared by. For nstance, the scalar multplcatve dentty s the scalar and the addtve dentty n s,,,...,. The standard bass for s smply whch s the standard bass for R n. Because ths bass contans n vectors, t follows that the dmenson of s n. Other bases exst; n fact, any lnearly ndependent set of n vectors n can be used, as demonstrated n Example 2. e,,,..., e 2,,,...,.. e n,,,..., EXAMPLE 2 Verfyng a Bass Show that S,,,,,,,, s a bass for C 3. Soluton Because C 3 has a dmenson of 3, the set v, v 2, v 3 wll be a bass f t s lnearly ndependent. To check for lnear ndependence, set a lnear combnaton of the vectors n S equal to as follows. c,, c 2, c 2,,, c 3,, Ths mples that c c 2 v v 2 v 3 c 2 c 3. c v c 2 v 2 c 3 v 3,, c c 2, c 2, c 3,, So, c c 2 c 3, and you can conclude that v, v 2, v 3 s lnearly ndependent. EXAMPLE 3 Representng a Vector n by a Bass Use the bass S n Example 2 to represent the vector v 2,, 2.

3 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 495 Soluton By wrtng you can obtan whch mples that c 2 and So, v c v c 2 v 2 c 3 v 3 c c 2, c 2, c 3 2,, 2, c c 2 2 c 2 c 2 c and c 3 2 v 2 v v 2 2 v Try verfyng that ths lnear combnaton yelds 2,, 2. Other than, there are several addtonal examples of complex vector spaces. For nstance, the set of m n complex matrces wth matrx addton and scalar multplcaton forms a complex vector space. Example 4 descrbes a complex vector space n whch the vectors are functons. EXAMPLE 4 The Space of Complex-Valued Functons Consder the set S of complex-valued functons of the form f x) f x f 2 x where f and f 2 are real-valued functons of a real varable. The set of complex numbers form the scalars for S and vector addton s defned by f x g x f x f 2 x g (x g 2 x f x g x f 2 x g 2 x. It can be shown that S, scalar multplcaton, and vector addton form a complex vector space. For nstance, to show that S s closed under scalar multplcaton, let c a b be a complex number. Then cf x a b f x f 2 x af x bf 2 x bf x af 2 x s n S.

4 496 CHAPTER 8 COMPLEX VECTOR SPACES The defnton of the Eucldean nner product n s smlar to that of the standard dot product n R n, except that here the second factor n each term s a complex conjugate. Defnton of Eucldean Inner Product n Let u and v be vectors n. The Eucldean nner product of u and v s gven by u v u v u 2 v 2 u n v n. REMARK: Note that f u and v happen to be real, then ths defnton agrees wth the standard nner (or dot) product n. R n EXAMPLE 5 Fndng the Eucldean Inner Product n C 3 Determne the Eucldean nner product of the vectors u 2,, 4 5 and v, 2,. Soluton u v u v u 2 v 2 u 3 v Several propertes of the Eucldean nner product are stated n the followng theorem. Theorem 8.7 Propertes of the Eucldean Inner Product Let u, v, and w be vectors n propertes are true u v v u u v w u w v w ku v k u v u kv k u v u u 6. u u f and only f u. and let k be a complex number. Then the followng Proof The proof of the frst property s gven, and the proofs of the remanng propertes have been left to you. Let u u, u 2,..., u n and v v, v 2,..., v n. Then v u v u v 2 u 2... v n u n v u v 2 u 2... v n u n v u v 2 u 2... v n u n

5 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 497 u v u 2 v 2 u v. u n v n You wll now use the Eucldean nner product n to defne the Eucldean norm (or length) of a vector n and the Eucldean dstance between two vectors n. Defnton of Norm The Eucldean norm (or length) of u n s denoted by u and s and Dstance n u u u 2. The Eucldean dstance between u and v s d u, v u v. The Eucldean norm and dstance may be expressed n terms of components as u u 2 u u n 2 2 d u, v u v 2 u 2 v u n v n 2 2. EXAMPLE 6 Fndng the Eucldean Norm and Dstance n Determne the norms of the vectors u 2,, 4 5 and v, 2, and fnd the dstance between u and v. Soluton The norms of u and v are gven as follows. u u 2 u 2 2 u v v 2 v 2 2 v The dstance between u and v s gven by d u, v u v, 2,

6 498 CHAPTER 8 COMPLEX VECTOR SPACES Complex Inner Product Spaces The Eucldean nner product s the most commonly used nner product n. However, on occason t s useful to consder other nner products. To generalze the noton of an nner product, use the propertes lsted n Theorem 8.7. Defnton of a Complex Inner Product Let u and v be vectors n a complex vector space. A functon that assocates wth u and v the complex number u, v s called a complex nner product f t satsfes the followng propertes.. u, v v, u 2. u v, w u, w v, w 3. ku, v k u, v 4. u, u and u, u f and only f u. A complex vector space wth a complex nner product s called a complex nner product space or untary space. EXAMPLE 7 A Complex Inner Product Space Let u u and be vectors n the complex space C 2, u 2 v v, v 2. Show that the functon defned by u, v u v 2u 2 v 2 s a complex nner product. Soluton Verfy the four propertes of a complex nner product as follows.. v, u v u 2v 2 u 2 u v 2u 2 v 2 u, v 2. u v, w u v w 2 u 2 v 2 w 2 u w 2u 2 w 2 v w 2v 2 w 2 u, w v, w 3. ku, v ku v 2 ku 2 v 2 k u v 2u 2 v 2 k u, v 4. u, u u u 2u 2 u 2 u 2 2 u 2 2 Moreover, u, u f and only f u u 2. Because all propertes hold, u, v s a complex nner product.

7 SECTION 8.4 EXERCISES 499 SECTION 8.4 EXERCISES In Exercses 8, perform the ndcated operaton usng u, 3, v 2, 3, and w 4, 6.. 3u 2. 4w 3. 2 w 4. v 3w 5. u 2 v v 2 2 w 7. u v 2w 8. 2v 3 w u In Exercses 9 2, determne whether S s a bass for. 9. S,,,. S,,,. S,,,,,,,, 2. S,,, 2,,,,, In Exercses 3 6, express v as a lnear combnaton of each of the bass vectors. (a),,,,,,,, (b),,,,,,,, 3. v, 2, 4. v,, 3 5. v, 2, 6. v,, In Exercses 7 24, determne the Eucldean norm of v. 7. v, 8. v, 9. v 3 6, 2 2. v 2 3, v, 2, 22. v,, 23. v 2,, 3, 24. v 2,, 2, 4 In Exercses 25 3, determne the Eucldean dstance between u and v u,, v, u 2, 4,, v 2, 4, u, 2, 3, v,, u 2, 2,, v,, u,, v, u, 2,, 2, v, 2,, 2 In Exercses 3 34, determne whether the set of vectors s lnearly ndependent or lnearly dependent. 3.,,, 32.,,,,,, 2,, 33.,,,,,,,, 34.,,,,,,,, In Exercses 35 38, determne whether the functon s a complex nner product, where u u, u 2 and v v, v u, v 4u v 6u 2 v u, v u v u 2 v Let v,, and v 2,,. If v 3 z, z 2, z 3 and the set v s not a bass for C 3, v 2, v 3, what does ths mply about z, z 2, and z 3? 4. Let v,, and v 2,,. Determne a vector v 3 such that v s a bass for C 3, v 2, v 3. In Exercses 4 45, prove the gven property where u, v, and w are vectors n and k s a complex number. 4. u v w u w v w u kv k u v 44. u u 45. u u f and only f u. 46. Wrtng Let u, v be a complex nner product and k a complex number. How are u, v and u, kv related? In Exercses 47 and 48, determne the lnear transformaton T : C m that has the gven characterstcs In Exercses 49 52, the lnear transformaton T : C m s gven by T v Av. Fnd the mage of v and the premage of w. 49. A 5. A u, v u u 2 v 2 u, v u v 2 u 2 v 2 T, 2,, T,, T, ) 2,, T,, A A,, v,, v 2 3 2,, v, v 2 5, w w w ku v k u v w 53. Fnd the kernel of the lnear transformaton gven n Exercse Fnd the kernel of the lnear transformaton gven n Exercse 5.

8 5 CHAPTER 8 COMPLEX VECTOR SPACES In Exercses 55 and 56, fnd the mage of v, for the ndcated 58. Determne whch of the followng sets are subspaces of the composton, where T and T 2 are gven by the followng matrces. vector space of complex-valued functons (see Example 4). and A 2 A (a) The set of all functons f satsfyng f. (b) The set of all functons f satsfyng f. 55. T 2 T (c) The set of all functons f satsfyng f f. 56. T T Determne whch of the followng sets are subspaces of the vector space of 2 2 complex matrces. (a) The set of 2 2 symmetrc matrces. (b) The set of 2 2 matrces A satsfyng A T A. (c) The set of 2 2 matrces n whch all entres are real. (d) The set of 2 2 dagonal matrces.

332600_08_1.qxp 4/17/08 11:29 AM Page 481

336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

2.3 Nilpotent endomorphisms

s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra

MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A

More information

Homework Notes Week 7

Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons

More information

Formulas for the Determinant

page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

2 More examples with details

Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and

More information

Composite Hypotheses testing

Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Some basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C

Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +

More information

NOTES ON SIMPLIFICATION OF MATRICES

NOTES ON SIMPLIFICATION OF MATRICES JONATHAN LUK These notes dscuss how to smplfy an (n n) matrx In partcular, we expand on some of the materal from the textbook (wth some repetton) Part of the exposton

More information

SL n (F ) Equals its Own Derived Group

Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu

More information

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS

BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all

More information

Orthogonal Functions and Fourier Series. University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don Fussell

Orthogonal Functons and Fourer Seres Vector Spaces Set of ectors Closed under the followng operatons Vector addton: 1 + 2 = 3 Scalar multplcaton: s 1 = 2 Lnear combnatons: Scalars come from some feld F

More information

w ). Then use the Cauchy-Schwartz inequality ( v w v w ).] = in R 4. Can you find a vector u 4 in R 4 such that the

Math S-b Summer 8 Homework #5 Problems due Wed, July 8: Secton 5: Gve an algebrac proof for the trangle nequalty v+ w v + w Draw a sketch [Hnt: Expand v+ w ( v+ w) ( v+ w ) hen use the Cauchy-Schwartz

More information

5 The Rational Canonical Form

5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

Poisson brackets and canonical transformations

rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

COMPUTING THE NORM OF A MATRIX

COMPUTING THE NORM OF A MATRIX KEITH CONRAD 1. Introducton In R n there s a standard noton of length: the sze of a vector v = (a 1,..., a n ) s v = a 2 1 + + a2 n. We wll dscuss n Secton 2 the general

More information

Vector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.

Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm

More information

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,

The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There

More information

p 1 c 2 + p 2 c 2 + p 3 c p m c 2

Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance

More information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998

More information

Orthogonal Functions and Fourier Series. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Orthogonal Functons and Fourer Seres Fall 21 Don Fussell Vector Spaces Set of ectors Closed under the followng operatons Vector addton: 1 + 2 = 3 Scalar multplcaton: s 1 = 2 Lnear combnatons: Scalars come

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

More metrics on cartesian products

More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Report on Image warping

Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

7. Products and matrix elements

7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ

More information

Grover s Algorithm + Quantum Zeno Effect + Vaidman

Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the

More information

Math 594. Solutions 1

Math 594. Solutons 1 1. Let V and W be fnte-dmensonal vector spaces over a feld F. Let G = GL(V ) and H = GL(W ) be the assocated general lnear groups. Let X denote the vector space Hom F (V, W ) of lnear

More information

CHAPTER III Neural Networks as Associative Memory

CHAPTER III Neural Networs as Assocatve Memory Introducton One of the prmary functons of the bran s assocatve memory. We assocate the faces wth names, letters wth sounds, or we can recognze the people

More information

Cocyclic Butson Hadamard matrices and Codes over Z n via the Trace Map

Contemporary Mathematcs Cocyclc Butson Hadamard matrces and Codes over Z n va the Trace Map N. Pnnawala and A. Rao Abstract. Over the past couple of years trace maps over Galos felds and Galos rngs have

More information

Lecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.

prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

Advanced Quantum Mechanics

Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

1 Vectors over the complex numbers

Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

Quantum Mechanics I - Session 4

Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................

More information

CHAPTER 4. Vector Spaces

man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear

More information

Eigenvalues of Random Graphs

Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

Ballot Paths Avoiding Depth Zero Patterns

Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,

More information

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization

DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.

More information

Random Walks on Digraphs

Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

More information

Lecture 2: Gram-Schmidt Vectors and the LLL Algorithm

NYU, Fall 2016 Lattces Mn Course Lecture 2: Gram-Schmdt Vectors and the LLL Algorthm Lecturer: Noah Stephens-Davdowtz 2.1 The Shortest Vector Problem In our last lecture, we consdered short solutons to

More information

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed

More information

Refined Coding Bounds for Network Error Correction

Refned Codng Bounds for Network Error Correcton Shenghao Yang Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, N.T., Hong Kong shyang5@e.cuhk.edu.hk Raymond W. Yeung Department

More information

MATH Homework #2

MATH609-601 Homework #2 September 27, 2012 1. Problems Ths contans a set of possble solutons to all problems of HW-2. Be vglant snce typos are possble (and nevtable). (1) Problem 1 (20 pts) For a matrx

More information

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac

More information

Matrix Approximation via Sampling, Subspace Embedding. 1 Solving Linear Systems Using SVD

Matrx Approxmaton va Samplng, Subspace Embeddng Lecturer: Anup Rao Scrbe: Rashth Sharma, Peng Zhang 0/01/016 1 Solvng Lnear Systems Usng SVD Two applcatons of SVD have been covered so far. Today we loo

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs

More information

LECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.

LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,

More information

Restricted Lie Algebras. Jared Warner

Restrcted Le Algebras Jared Warner 1. Defntons and Examples Defnton 1.1. Let k be a feld of characterstc p. A restrcted Le algebra (g, ( ) [p] ) s a Le algebra g over k and a map ( ) [p] : g g called

More information

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values

Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly

More information

ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013

ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run

More information

Affine transformations and convexity

Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Games of Threats. Elon Kohlberg Abraham Neyman. Working Paper

Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017

More information

Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2

P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons

More information

Supplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso

Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

The exponential map of GL(N)

The exponental map of GLN arxv:hep-th/9604049v 9 Apr 996 Alexander Laufer Department of physcs Unversty of Konstanz P.O. 5560 M 678 78434 KONSTANZ Aprl 9, 996 Abstract A fnte expanson of the exponental

More information

arxiv: v1 [quant-ph] 6 Sep 2007

An Explct Constructon of Quantum Expanders Avraham Ben-Aroya Oded Schwartz Amnon Ta-Shma arxv:0709.0911v1 [quant-ph] 6 Sep 2007 Abstract Quantum expanders are a natural generalzaton of classcal expanders.

More information

9 Characteristic classes

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct

More information

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of

Chapter 7 Generalzed and Weghted Least Squares Estmaton The usual lnear regresson model assumes that all the random error components are dentcally and ndependently dstrbuted wth constant varance. When

More information

MTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i

MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that

More information

Problem Set 9 Solutions

Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

INTERVAL SEMIGROUPS. W. B. Vasantha Kandasamy Florentin Smarandache

Interval Semgroups - Cover.pdf:Layout 1 1/20/2011 10:04 AM Page 1 INTERVAL SEMIGROUPS W. B. Vasantha Kandasamy Florentn Smarandache KAPPA & OMEGA Glendale 2011 Ths book can be ordered n a paper bound reprnt

More information

Which Separator? Spring 1

Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w \$ + b) proportonal

More information

The Geometry of Logit and Probit

The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Inexact Newton Methods for Inverse Eigenvalue Problems

Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.

More information

Feb 14: Spatial analysis of data fields

Feb 4: Spatal analyss of data felds Mappng rregularly sampled data onto a regular grd Many analyss technques for geophyscal data requre the data be located at regular ntervals n space and/or tme. hs s

More information

2. Differentiable Manifolds and Tensors

. Dfferentable Manfolds and Tensors.1. Defnton of a Manfold.. The Sphere as a Manfold.3. Other Examples of Manfolds.4. Global Consderatons.5. Curves.6. Functons on M.7. Vectors and Vector Felds.8. Bass

More information

Mathematical Preparations

1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

On the set of natural numbers

On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers

More information

Determinants Containing Powers of Generalized Fibonacci Numbers

1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal

More information

An efficient algorithm for multivariate Maclaurin Newton transformation

Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,

More information

be a second-order and mean-value-zero vector-valued process, i.e., for t E

CONFERENCE REPORT 617 DSCUSSON OF TWO PROCEDURES FOR EXPANDNG A VECTOR-VALUED STOCHASTC PROCESS N AN ORTHONORMAL WAY by R. GUTkRREZ and M. J. VALDERRAMA 1. ntroducton Snce K. Karhunen [l] and M. Lo&e [2]

More information

Subset Topological Spaces and Kakutani s Theorem

MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered

More information

Errors for Linear Systems

Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons Â and ˆb avalable. Then the best thng we can do s to solve Âˆx ˆb exactly whch

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

arxiv: v2 [quant-ph] 29 Jun 2018

Herarchy of Spn Operators, Quantum Gates, Entanglement, Tensor Product and Egenvalues Wll-Hans Steeb and Yorck Hardy arxv:59.7955v [quant-ph] 9 Jun 8 Internatonal School for Scentfc Computng, Unversty

More information

Online Classification: Perceptron and Winnow

E0 370 Statstcal Learnng Theory Lecture 18 Nov 8, 011 Onlne Classfcaton: Perceptron and Wnnow Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton In ths lecture we wll start to study the onlne learnng

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Dynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)

/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes

More information

Polynomial Identities of RA2 Loop Algebras

Ž. Journal of Algebra 213, 557566 1999 Artcle ID jabr.1998.7675, avalable onlne at http:www.dealbrary.com on Polynomal Identtes of RA2 Loop Algebras S. O. Juraans and L. A. Peres Departamento de Matematca,

More information