Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal


 Harold Howard
 4 years ago
 Views:
Transcription
1 Inner Product Defnton 1 () A Eucldean space s a fntedmensonal 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, postvedefnte functon, : X X R (x,x) x,x. (Postvedefnte means x, x > 0 unless x = 0.) 1 2 Orthogonal Defnton 3 (Orthogonal) Two vectors x and x are orthogonal f ther nner product s zero, x,x =0. Geometrcally, orthogonal means perpendcular. Orthonormal Bass Defnton 4 (Orthonormal Bass) In a Eucldean space, an orthonormal bass s a bass x such that x,x j = Any two bass vectors are orthogonal. 1 f = j 0 f j. A Eucldean space has more than one orthonormal bass. 3 4 If then x= x x x = x x, x,x = x x. For the real numbers R, the nner product s just ordnary multplcaton. R Defnton 5 The Eucldean space R of real numbers s defned by the nner product x,x := x x. 5 6
2 R n The Eucldean space R n := R R (n tmes), n whch the elements are vectors wth n real components. By assumpton, the n vectors ,, form an orthonormal bass. The nner product of two vectors s then the sum of the component by component products. 7 Isomorphc In abstract algebra, somorphc means the same. If two objects of a gven type (group, rng, vector space, Eucldean space, algebra, etc.) are somorphc, then they are the same, when consdered as objects of that type. An somorphsm s a onetoone and onto mappng from one space to the other that preserves all propertes defnng the space. Any ndmensonal Eucldean space s somorphc to R n. Although two spaces may be somorphc as Eucldean spaces, perhaps the same two spaces are not somorphc when vewed as another space. 8 CoordnateFree Versus Bass It s useful to thnk of a vector n a Eucldean space as coordnatefree. Gven a bass, any vector can be expressed unquely as a lnear combnaton of the bass elements. For example, f x= x x for some bass x, one can refer to the x as the coordnates of x n terms of ths bass. Many lnear algebra textbooks develop all the results n terms of a bass. In economc theory and econometrcs, typcally vectors are not seen as coordnatefree. A partcular bass s sngled out, and one works wth coordnates. Commonly there s a natural bass, but unfortunately the natural bass s perhaps not orthonormal. Despte ths tradton, the coordnatefree pontofvew s superor. Not usng coordnates reduces the use of subscrpts and makes expressons smpler, and theorems are easer to state and to prove Lnear Transformaton Defnton 6 (Lnear Transformaton) A lnear transformaton from a Eucldean space X to a Eucldean space Y s a functon such that A : X Y x y=ax A(x 1 + x 2 )=Ax 1 + Ax Adjont The followng proposton s a standard theorem of lnear algebra. Proposton 7 (Adjont) Gven a lnear transformaton A : X Y, then there exsts a unque lnear transformaton (the adjont) A : Y X that preserves the nner product: y,ax = A y,x for all x and y. 12 (1)
3 The adjont s very mportant n applcatons and has not been apprecated by economsts. The adjont s ndependent of any choce of bases, and n many applcatons one can determne t drectly, expressed n a coordnatefree way. The adjont then becomes a powerful tool, and one can easly obtan valuable results va the adjont, almost as f by magc. Typcally one does not calculate the adjont drectly. Instead one conjectures an expresson for the adjont, and then verfes that the adjont condton (1) holds. Matrx Representaton A matrx representaton for a lnear transformaton A : X Y s a matrx A j that shows how bass elements x j X map to a lnear combnaton of bass elements y Y: x j Ax j = A j y Adjont as Transpose If the bases for X and Y are each orthonormal, then the matrx representaton of the adjont s the transpose of the matrx representaton: A y = A j x j. j To prove ths relatonshp, verfy the adjont condton (1), for arbtrary bass elements: A y,x j = A k x k,x j k = A j (snce the bass x j s orthonormal) = y, A k j y k (snce the bass y s orthonormal) k = y,ax j, as desred On the other hand, f the bases are not orthonormal, then the transpose of the matrx representaton s not the matrx representaton of the adjont. Snce we want to see vectors as coordnatefree, however, the matrx representaton s of secondary mportance. Apart from smple cases, t may be dffcult to wrte down the matrx representaton explctly. At the same tme, one can descrbe the adjont easly, wthout reference to any bass
4 For some y X, the adjont of the lnear functon Resz Representaton A fundamental theorem states that any lnear functon X R can be expressed as x y,x for a unque y. s y : R X z x=zy y : X R x z= y,x Verfy that the adjont condton (1) holds: Thus x,yz = x,y z= y,x z= y,x,z = y x= y,x. y x,z. Ether notaton s equvalent, but normally we employ the nner product notaton on the rghthand sde. 21 Matrx Representaton Suppose y= y x, for a bass x. Let us use the natural orthonormal bass 1 for R. The matrx representaton of the lnear transformaton y s 1 y x, so the vector wth components y defnes the matrx representaton. For the adjont y, however, the matrx representaton s not the transpose of ths vector, unless the bass x s orthonormal. 22 The matrx representaton of the adjont s x j y,x j 1= y x,x j 1= x,x j y 1. For a nonorthonormal bass, the matrx representaton of the adjont s not x j y j 1. Fundamental Theorem of Lnear Algebra The fundamental theorem of lnear algebra states that the null space N(A) and the range R ( A ) are orthogonal, and any x X can be wrtten unquely as an element of N(A) plus an element of R ( A ). The same relatonshp holds for the range R(A) and the null space N ( A )
5 MoorePenrose Generalzed Inverse Usng the fundamental theorem of lnear algebra, we defne the MoorePenrose generalzed nverse. Consder a lnear transformaton A : X Y x y=ax. The generalzed nverse A + s a lnear transformaton mappng Y X. Usng the fundamental theorem of lnear algebra, one can prove the followng. Proposton 8 (Inverse) The restrcton of A to R ( A ) ( A : R A ) R(A) x y=ax. s onetoone and onto, so t has an nverse Defne the generalzed nverse A + : Y X va ths nverse mappng. For y R(A), defne A + y as the nverse of A. For y N ( A ), defne A + y=0. Ths defnton rests on the coordnatefree approach to lnear algebra. The relatonshp between A and A + s symmetrc. A lnear transformaton s the generalzed nverse of ts generalzed nverse: (A + ) + = A. And AA + A=A. The sngularvalue decomposton obtans further results. If A s onto, then AA s nvertble, and A + = A (AA ) For the lnear equaton Lnear Equaton Ax=y, there s a soluton x f and only f y R(A). If there s a soluton, then the unque soluton n R ( A ) s A + y. Ths vector plus any element of N(A) s also a soluton. Hence the complete soluton set s { A + y } + N(A). 28 Defnton 9 (Partal Orderng) Gven a bass representaton Order The concept of a Eucldean space does not nvolve any concept of order, of one vector beng greater than another. However commonly one defnes the addtonal structure of a partal orderng va the representaton of a vector n a natural bass. then x= x j x j, j x 0 f every x j 0 x 0 f every x j 0 and some x > 0 x 0 f every x j >
6 Defnton of the Inner Product To embed model structure nto the nner product smplfes the analyss. In a Eucldean space of random varables, one mght defne the nner product of two random varables as the covarance. Orthogonalty then means no correlaton. A dfferent defnton of the nner product derves from a partal orderng: one defnes a trace nner product consstent wth the orderng. 31
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 informationAPPENDIX 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 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 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 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 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 information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3d rotaton around the xaxs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3d rotaton around the yaxs:
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 informationSalmon: Lectures on partial differential equations. Consider the general linear, secondorder PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of secondorder equatons There are general methods for classfyng hgherorder partal dfferental equatons. One s very general (applyng even to
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 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 Durgapur713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
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 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 informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr GyeSeon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More information3.1 Expectation of Functions of Several Random Variables. )' be a kdimensional 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 informationFINITELYGENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELYGENERTED 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 fntelygenerated module. (1) There
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 informationw ). Then use the CauchySchwartz inequality ( v w v w ).] = in R 4. Can you find a vector u 4 in R 4 such that the
Math Sb 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 CauchySchwartz
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 informationDeriving the XZ Identity from Auxiliary Space Method
Dervng the XZ Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCCThe Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationMathematical 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 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 informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
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 informationTimeVarying Systems and Computations Lecture 6
TmeVaryng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationRepresentation 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 informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The nonrelativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The nonrelatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
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 informationReport 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 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 nonlnear case, and also
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 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 informationFeb 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 informationHowever, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigenvalues
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 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 informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 50757 HIKARI Ltd, wwwmhkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationComposite 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 informationCoordinateFree Projective Geometry for Computer Vision
MM Research Preprnts,131 165 No. 18, Dec. 1999. Beng 131 CoordnateFree Proectve Geometry for Computer Vson Hongbo L, Gerald Sommer 1. Introducton How to represent an mage pont algebracally? Gven a Cartesan
More informationMath 396. Metric tensor on hypersurfaces
Math 396. Metrc tensor on hypersurfaces 1. Motvaton Let U R n be a nonempty open subset and f : U R a C functon. Let Γ U R be the graph of f. The closed subset Γ n U R proects homeomorphcally onto U
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 informationR n α. . The funny symbol indicates DISJOINT union. Define an equivalence relation on this disjoint union by declaring v α R n α, and v β R n β
Readng. Ch. 3 of Lee. Warner. M s an abstract manfold. We have defned the tangent space to M va curves. We are gong to gve two other defntons. All three are used n the subject and one freely swtches back
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
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 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 informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 OO 4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 OO 5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More information17. CoordinateFree Projective Geometry for Computer Vision
17. CoordnateFree Projectve Geometry for Computer Vson Hongbo L and Gerald Sommer Insttute of Computer Scence and Appled Mathematcs, ChrstanAlbrechtsUnversty of Kel 17.1 Introducton How to represent
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 informationALGEBRA MIDTERM. 1 Suppose I is a principal ideal of the integral domain R. Prove that the Rmodule I R I has no nonzero torsion elements.
ALGEBRA MIDTERM CLAY SHONKWILER 1 Suppose I s a prncpal deal of the ntegral doman R. Prove that the Rmodule I R I has no nonzero torson elements. Proof. Note, frst, that f I R I has no nonzero torson
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
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 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 semdefnte for all
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
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 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 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 informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 111 Emprcal Models 112 Smple Lnear Regresson 113 Propertes of the Least Squares Estmators 114 Hypothess Test n Smple Lnear Regresson 114.1 Use of ttests
More informationU.C. Berkeley CS294: Beyond WorstCase Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond WorstCase 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 informationDifferential Polynomials
JASS 07  Polynomals: Ther Power and How to Use Them Dfferental Polynomals Stephan Rtscher March 18, 2007 Abstract Ths artcle gves an bref ntroducton nto dfferental polynomals, deals and manfolds and ther
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 faketest data; fxed
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationChapter 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 informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECONDORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECONDORDER HYPERBOLIC EUATION
More informationThe Second AntiMathima on Game Theory
The Second AntMathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2player 2acton zerosum games 2. 2player
More information(A and B must have the same dmensons to be able to add them together.) Addton s commutatve and assocatve, just lke regular addton. A matrx A multpled
CNS 185: A Bref Revew of Lnear Algebra An understandng of lnear algebra s crtcal as a steppngo pont for understandng neural networks. Ths handout ncludes basc dentons, then quckly progresses to elementary
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 Wrapup In whch we talk about even more generalzatons of Cheeger s nequaltes, and
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 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 information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the righthand 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 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 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 informationAdvanced 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 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 informationOn the symmetric character of the thermal conductivity tensor
On the symmetrc character of the thermal conductvty tensor Al R. Hadjesfandar Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New York Buffalo, NY 146 USA ah@buffalo.edu
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationError Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
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 informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. > Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More information15.1 The geometric basis for the trifocal tensor Incidence relations for lines.
15 The Trfocal Tensor The trfocal tensor plays an analogous role n three vews to that played by the fundamental matrx n two. It encapsulates all the (projectve) geometrc relatons between three vews that
More informationFrom BiotSavart Law to Divergence of B (1)
From BotSavart Law to Dvergence of B (1) Let s prove that BotSavart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of BotSavart. The dervatve s wth respect to
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationSignal space Review on vector space Linear independence Metric space and norm Inner product
Sgnal space.... Revew on vector space.... Lnear ndependence... 3.3 Metrc space and norm... 4.4 Inner product... 5.5 Orthonormal bass... 7.6 Waveform communcaton system... 9.7 Some examples... 6 Sgnal space
More informationLorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3
Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationSystems of Equations (SUR, GMM, and 3SLS)
Lecture otes on Advanced Econometrcs Takash Yamano Fall Semester 4 Lecture 4: Sstems of Equatons (SUR, MM, and 3SLS) Seemngl Unrelated Regresson (SUR) Model Consder a set of lnear equatons: $ + ɛ $ + ɛ
More informationThe Finite Element Method: A Short Introduction
Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental
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 informationChat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980
MT07: Multvarate Statstcal Methods Mke Tso: emal mke.tso@manchester.ac.uk Webpage for notes: http://www.maths.manchester.ac.uk/~mkt/new_teachng.htm. Introducton to multvarate data. Books Chat eld, C. and
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06FnteDfference Method (Onedmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More information10801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
080: 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 informationPerron 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