Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
|
|
- Rudolph Bryant
- 5 years ago
- Views:
Transcription
1 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 choose our norms analogous to the way we dd for vector norms; e.g., we could assocate the number max j a j. However, ths s actually not very useful because remember our goal s to study lnear systems A x = b. The general defnton of a matrx norm s a map from all m n matrces to IR 1 whch satsfes certan propertes. However, the most useful matrx norms are those that are generated by a vector norm; agan the reason for ths s that we want to solve A x = b so f we take the norm of both sdes of the equaton t s a vector norm and on the left hand sde we have the norm of a matrx tmes a vector. We wll defne an nduced matrx norm as the largest amount any vector s magnfed when multpled by that matrx,.e., A = max A x Note that all norms on the rght hand sde are vector norms. We wll denote a vector and matrx norm usng the same notaton; the dfference should be clear from the argument. We say that the vector norm on the rght hand sde nduces the matrx norm on the left. Note that sometmes the defnton s wrtten n an equvalent way as A = sup A x What s I? Clearly t s just one. The problem wth the defnton s that t doesn t tell us how to compute a matrx norm for a general matrx A. The followng theorem gves us a way to calculate matrx norms nduced by the l and l 1 norms; the matrx norm nduced by l 2 norm wll be addressed later after we have ntroduced egenvalues. Theorem Let A be an m n matrx. Then A = max 1 m A 1 = max 1 j n [ m [ n =1 ] a j ] a j (max absolute row sum) (max absolute column sum) 1
2 Determne A and A 1 where A = We have A 1 = max{( ), (2 + 1), ( )} = max{24, 3, 18} = 24 A = max{( ), (3 + 12), ( )} = max{7, 15, 23} = 23 Proof We wll prove that A s the maxmum row sum (n absolute value). We wll do ths by provng that A max 1 m [ n Frst recall that f A x = b then b = ] a j and then showng a j x j b = max For the frst nequalty we know that by defnton Now lets smplfy the numerator to get A x = max A = max a j x j max b = max A x A max 1 m a j x j a j x j max [ n a j ] a j Thus the rato reduces to A x n max a j = max a j and hence A = max A x Now for the second nequalty we know that max a j. A A y y 2
3 for any y IR n because equalty n the defnton holds here for the maxmum of ths rato. So now we wll choose a partcular y and we wll construct t so that t has y = 1. Frst let p be the row where A has ts maxmum row sum (or there are two rows, take the frst),.e., max a j = a pj Now we wll take the entres of y to be ±1 so ts nfnty norm s one. Specfcally we choose y = { 1 f apj 0 1 f a pj < 0 Defnng y n ths way means that a j y j = a j. Usng ths and the fact that y = 1 we have A A y y = max a j y j a pj y j = n a pj = a pj but the last quantty on the rght s must the maxmum row sum and the proof s complete. What are the propertes that any matrx norm (and thus an nduced norm) satsfy? You wll recognze most of them as beng analogous to the propertes of a vector norm. () A 0 and = 0 only f a j = 0 for all, j. () ka = k A for scalars k () AB A B (v) A + B A + B A very useful nequalty s A x A for any nduced norm Why s ths true? A = max A x A x A x A The Condton Number of a Matrx We sad one of our goals was to determne f small changes n our data of a lnear system produces small changes n the soluton. Now lets assume we want to solve A x = b, b 0 but nstead we solve A y = b + δb that s, we have perturbed the rght hand sde by a small about δb. We assume that A s nvertble,.e., A 1 exsts. For smplcty, we have not perturbed the coeffcent matrx 3
4 A. What we want to see s how much y dffers from x. Lets wrte y as x + δx and so our change n the soluton wll be δx. The two systems are A x = b A( x + δx) = b + δb What we would lke to get s an estmate for the relatve change n the soluton,.e., δx n terms of the relatve change n b where denotes any nduced vector norm. Subtractng these two equatons gves A δx = δb whch mples δx = A 1 δb Now we take the (vector) norm of both sdes of the equaton and then use our favorte nequalty above δx = A 1 δb A 1 δb Remember our goal s to get an estmate for the relatve change n the soluton so we have a bound for the change. What we need s a bound for the relatve change. Because A x = b we have A x = b b A 1 b 1 A Now we see that f we dvde our prevous result for δx by A > 0 we can use ths result to ntroduce b n the denomnator. We have δx A A 1 δb A Multplyng by A gves the desred result A 1 δb b δx A A 1 δb b If the quantty A A 1 s small, then ths means small relatve changes n b result n small relatve changes n the soluton but f t s large, we could have a large relatve change n the soluton. Defnton The condton number of a square matrx A s defned as K(A) A A 1 Note that the condton number depends on what norm you are usng. We say that a matrx s well-condtoned f K(A) s small and ll-condtoned otherwse. 4
5 Fnd the condton number of the dentty matrx usng the nfnty norm. Clearly t s one because the nverse of the dentty matrx s tself. Fnd the condton number for each of the followng matrces usng the nfnty norm.. Explan geometrcally why you thnk ths s happenng. A 1 = A = Frst we need to fnd the nverse of each matrx and then take the norms. Note the followng trck for takng the nverse of a 2 2 matrx A 1 1 = A = Now K (A 1 ) = (7)(1) = 7 K (A 2 ) = (3)(1000) = 3000 Calculate the K (A) where A s a b A = c d and comment on when the condton number wll be large. A 1 1 d b = ad bc c a So 1 1 K (A) = max{ a + b, c + d } max{ d + b, c + a } ad bc ad bc Consequently f the determnant ad bc 0 then the condton number can be qute large;.e., when the matrx s almost not nvertble,.e., almost sngular. A classc example of an ll condtoned matrx s the Hlbert matrx whch we have already encountered Here are some of the condton numbers (usng the matrx norm nduced by the l 2 vector norm). n approxmate K 2 (A) , ,
6 Vector (or Lnear) Spaces What we want to do now s take the concept of vectors and the propertes of Eucldean space IR n and generalze them to a collecton of objects wth two operatons defned, addton and scalar multplcaton. We want to defne these operatons so that the usual propertes hold,.e., x + y = y + x, k(x + y) = kx + ky for objects other than vectors, such as matrces, contnuous functons, etc. We want to be able to add two elements of our set and get another element of the set. Defnton A vector or lnear space V s a set of objects, whch we wll call vectors, for whch addton and scalar multplcaton are defned and satsfy the followng propertes. () x + y = y + x () x + (y + z) = (x + y) + z () there exsts a zero element 0 V such that x + 0 = 0 + x = x (v) for each x V there exsts x V such that x + ( x) = 0 (v) 1x = x (v) (α + β)x = αx + βx, for scalars α, β (v) α(x + y) = αx + αy (v) (αβ)x = α(βx) IR n s a vector space wth the usual operaton of addton and scalar multpl- caton. The set of all m n matrces wth the usual defnton of addton and scalar multplcaton forms a vector space. All polynomals of degree less than or equal to two on [ 1, 1] form a vector space wth usual defnton of addton and scalar multplcaton. Here p(x) = a 0 + a 1 x + a 2 x 2 All contnuous functons defned on [0, 1]. Note that f v, w V then v + w V ; we say that V s closed under addton. Also f v V and k s a scalar, kv V and t s closed under scalar multplcaton. Defnton A subspace S of a vector space V s a nonempty subset of V such that f we take any lnear combnaton of vectors n S t s also n S. A lne s a subspace of IR 2 because f we take add two ponts on the lne, the result s stll a pont on the lne and f we multply any pont on the lne by a constant then t s stll on the lne. matrces. All lower trangular matrces form a subspace of the vector space of all square 6
7 Is the set of all ponts on the gven lnes a subspace of IR 2? y = x y = 2x + 1 The frst forms a subspace but the second does not. It s not closed under addton or scalar multplcaton, e.g., 2(x, 2x + 1) = (2x, 4x + 2) whch s not on the lne y = 2x + 1 Lnear ndependence, spannng sets and bass We now want to nvestgate how to characterze a vector space, f possble. To do ths, we need the concepts of () lnear ndependence/dependence of vectors () a spannng set () the dmenson of a vector space (v) the bass of a fnte dmensonal vector space In IR 2 the vectors v 1 = (1, 0) T and v 2 = (0, 1) T are specal because () every other vector n IR 2 can be wrtten as a lnear combnaton of these two vectors, e.g., f x = (x 1, x 2 ) T then x = x 1 v 1 + x 2 v 2. () the vectors v 1, v 2 are dfferent n the sense that the only way we can combne them and get the zero vector s wth zero coeffcents,.e., C 1 v 1 + C 2 v 2 = 0 C 1 = C 2 = 0 Of course there are many other sets of two vectors n IR 2 that have the same propertes. Defnton Let V be a vector space. Then the set of vectors { v }, = 1,..., n () span V f any w V can be wrtten as a lnear combnaton of the v,.e., () are lnearly ndependent f w = C j v j C j v j = 0 C = 0, otherwse they are lnearly dependent. Note that ths says the only way we can combne the vectors and get the zero vector s f all coeffcents are zero. () form a bass for V f they are lnearly ndependent and span V. 7
8 Let V be the vector space of all polynomals of degree less than or equal to two on [0, 1]. Then the set of vectors (.e., polynomals) n V {1, x, x 2, 2 + 6x} span V but they are NOT lnearly ndependent because 2(1) + 6(x) + 0(x 2 ) (2 + 6x) = 0 and thus they do not form a bass for V. The set of vectors {1, x} are lnearly ndependent but they do not span V because, e.g., x 2 V and t can t be wrtten as a lnear combnaton of {1, x}. Defnton The number of elements n the bass for V s called the dmenson of V. It s the smallest number of vectors needed to completely characterze V,.e., that span V. Fnd a bass for () the vector space of all 2 2 matrces and () the vector space of all 2 2 symmetrc matrces. Gve the dmenson. A bass for all 2 2 matrces conssts of the four matrces 1 0, 0 1,, and so ts dmenson s 4. A bass for all 2 2 symmetrc matrces conssts of the three matrces 1 1,, and so ts dmenson s 2. Four Important Spaces 1. The column space (or equvalently the range) of A, where A s m n matrx s all lnear combnatons of the columns of A. We denote ths by R(A). By defnton t s a subspace of IR m. Consder the overdetermned system 3 4 x = b 1 b y b 3 We know that ths s solvable f there exsts x, y such that x y 2 4 = 6 b 1 b 2 b 3 whch says that b has to be n the column space (or range) of A. 8
9 Ths says that an equvalent statement to A x = b beng solvable s that b s n the range or column space of A. 2. The null space of A, denoted N (A), where A s m n matrx s the set of all vectors z IR n such that A z = 0. Fnd the null space of each matrx A 1 = 3 4 A 2 = 2 4 A 3 = For A 1 we have that N (A 1 ) = 0 because the matrx s nvertble. To see ths we could take the determnant or perform GE and get the result. For A 2 we see that N (A 2 ) s α( 2x 2, x 2 ) T,.e., the all ponts on the lne through the orgn y =.5x. To see ths consder GE for the system ( 2 4 ) ( Ths says that x 2 s arbtrary and x 1 = 2x 2. For A 3 we see that N (A 2 ) s all of IR 2. ) 0 x 2 = 0, x 1 + 2x 2 = 0 What are the possble null spaces of a 3 3 matrx? For an nvertble matrx t s the () zero vector n IR 3, we could have () a lne through the orgn, () a plane through the orgn or all of (v) IR 3. 9
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 information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 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
More informationn 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 informationInner 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 informationNorms, Condition Numbers, Eigenvalues and Eigenvectors
Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b
More informationErrors 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  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More information2.3 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= = = (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 informationU.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 informationChapter 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 informationMEM 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 information332600_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 information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationHomework 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 informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationMore 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 information3.1 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 informationQuantum 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 informationFormulas 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 informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationLinear, 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 informationVector 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 information2 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 informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationBOUNDEDNESS 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 informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationp 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 informationQuantum 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 informationThe 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 informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationMath 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 informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationCHAPTER 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 informationAffine 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 informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationThe 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 informationCOMPLEX 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 informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationSalmon: 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 informationPRIMES 2015 reading project: Problem set #3
PRIMES 2015 readng project: Problem set #3 page 1 PRIMES 2015 readng project: Problem set #3 posted 31 May 2015, to be submtted around 15 June 2015 Darj Grnberg The purpose of ths problem set s to replace
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More information5 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 informationExample: (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 informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationPoisson 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 informationComplex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)
Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For
More informationU.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 informationLinear Algebra and its Applications
Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationSL 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 informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationCHAPTER 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 informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationMATH Sensitivity of Eigenvalue Problems
MATH 537- Senstvty of Egenvalue Problems Prelmnares Let A be an n n matrx, and let λ be an egenvalue of A, correspondngly there are vectors x and y such that Ax = λx and y H A = λy H Then x s called A
More informationNOTES 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 informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationHowever, 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 informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede
Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationTransfer 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 informationLecture 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 informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationBallot 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 informationP A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that
Backward Error Analyss for House holder Reectors We want to show that multplcaton by householder reectors s backward stable. In partcular we wsh to show fl(p A) = P (A) = P (A + E where P = I 2vv T s the
More information1 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 informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More informationRandom 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 informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationStructure 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 informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More information20. 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 informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More information9 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 informationw ). 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 information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationComplex Numbers Alpha, Round 1 Test #123
Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus
More informationDISCRIMINANTS 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 informationFINITELY-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 informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationRestricted 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 informationC/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 informationLECTURE 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