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

Size: px
Start display at page:

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

Transcription

1 MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty

2 Outlne Vectors Matrces MATLAB Advanced Topcs

3 Vectors A vector s a one-dmensonal array of scalarelements (real or complex numbers) x1 x 2 column vector: x=, row vector: y = y1 y2 y xn Vectors of equal dmenson can be added (elementwse): x= a+ b x = a + b, = 1,, n 1 2 = Vectors can be multpled by a scalar: x= αa x = αa Vectors can be transposed: x x xn T [ x x x ] 1 2 n [ ] n

4 Inner Product & Norm The nner product of two n-dmensonal vectors x, y s xy, = xy, n = 1 for column vectors xy, = for row vectors xy, = xy The Eucldean norm or length of a vector x s x = x, x = x n = 1 2 T xy Other norms or length measures are also just useful: n x = x, x = max x 1 = 1 T

5 Lnear Combnatons of Vectors Suppose α, = 1,, p s a set of scalars and x, = 1,, p s a set of column or row vectors, then we defne a new vector y va the lnear combnaton: for columns y = α x + α x + + α x y1 x1,1 x p,1 = α α p yn x x , n p, n p p A set of p vectors x, = 1,, p s lnearly dependent f there exsts a nontrval set of constants α, = 1,, p such that α x + α x + + α x = 0 otherwse t s lnearly ndependent. p p A set of lnearly ndependent n-dmensonal vectors contans at most n vectors.

6 Matrces A matrx s a 2-dmensonal (rectangular) array of elements: m rows a11 a12 a1 n am1 amn n columns Sometmes we wrte A= aj The elements are called scalars, they are usually real or complex numbers. A matrx wth one row, m = 1, s called a row matrx or row vector. A matrx wth one column, n = 1, s called a column matrx or column vector.

7 Algebrac Operatons Equalty - two matrces (of the same sze) AB, are equal, wrtten B, f ther correspondng elements are equal, a = b, for1 m,1 j n j j Matrces of the same sze can be added and subtracted. Matrx A= addton and subtracton are performed element-wse A+ B= C a + b = c A B= C a b = c j j j j j j Any matrx A= aj can be multpled by a scalar α αa= αaj An m n matrx A can be post multpled by an n q matrx to produce an m q matrx C, = = C AB, c a b n j k= 1 k k B

8 Multplcaton b12 b 22 c 22 = a21 a22 a23 a 24 b23 b24 b12 b c a a a a a b [ ] = = k= 1 2k k2 b32 b 42

9 Transpose The transpose of an m n matrx s the n p matrx obtaned by nterchangng rows and columns: a11 am 1 a11 a12 a 1n a12 a m2 T A= A = am 1 am2 amn a1 n a mn T A matrx s symmetrc f A = A The followng rules obtan: ( AB) ( ) T T T = B A T T T A+ B = A + B

10 Determnant ( ) ( ) The j-th mnor M of a square n n matrx A s the n 1 n -1 submatrx of A obtaned by elmnatng row and column j. The determnant of a square matrx s defned recursvely. The determnant of a j= 1 j ( ) [ a ] 1 1 matrx s det = a The determnant of a n n matrx s defned by the expanson n det A= a γ for any = 1,2,, n j j where γ s the cofactor γ = 1 det M Note : 'for any ' means expand along any row. The same result s obtaned by expandng along any column. + j j j j

11 Propertes of Determnants multply any sngle row or column of A by scalar α to get A det A= α det A nterchange any two rows or columns of A to get A det A= det A add multple of any row or column to another row or column to get det A= det A T det A = det A, det AB= det Adet B for AC, square A B det det Adet D 0 D = for A nonsngular A B 1 det = det Adet D CA B C D A

12 Matrx Inverse An dentty matrx of sze n s the square matrx wth n rows and columns: I = The adjugate of a square matrx A s defned as the transpose of the matrx of cofactors: T adja = γ j It can be shown that AadjA= det A I ( ) adja If det A 0, we have A = I det A A square matrx A wth det A 0 s called nonsngular, for a nonsngular matrx we can defne the nverse adja = = = det A A AA I, A A I

13 Rank Consder an m n matrx A. The number of lnearly ndependent rows of A equals the number of ts lnearly ndependent columns. The rank of A s the number of ts lnearly ndependent rows or columns. ( m n) rank A max, If A s a square matrx of sze n, rank A= n det A 0

14 MATLAB Basc Operatons + Addton A and B must have the same sze, unless one s a scalar. A scalar can be added to a matrx of any sze. - Subtracton A and B must have the same sze, unless one s a scalar. A scalar can be subtracted from a matrx of any sze. * Matrx multplcaton. For nonscalar A and B, the number of columns of A must equal the number of rows of B. A scalar can multply a matrx of any sze. / Slash or matrx rght dvson. B/A s roughly the same as B*nv(A). More precsely, B/A = (A'\B')'. \ Backslash or matrx left dvson. If A s a square matrx, A\B s roughly the same as nv(a)*b, except t s computed n a dfferent way.

15 MATLAB Basc Operatons ^ Matrx power. X^p s X to the power p, f p s a scalar. If p s an nteger, the power s computed by repeated squarng. If the nteger s negatve, X s nverted frst. ' Matrx transpose. A' s the lnear algebrac transpose of A. For complex matrces, ths s the complex conjugate transpose..' Array transpose. A.' s the array transpose of A. For complex matrces, ths does not nvolve conjugaton. Note: The slash\backslash operatons are the better than nv to solve lnear equatons.

16 MATLAB Basc Functons norm rank det trace nv matrx or vector norm matrx rank determnant sum of the dagonal elements matrx nverse

17 Applcatons of Matrces Matrces are mportant n many applcatons. One of the most mportant s the soluton of sets of smultaneous lnear equatons: a x + a x + a x = b n n 1 a x + a x + a x = b n1 1 n2 2 nn n 2 =,f det 0 = 1 Ax b A x A b Another applcaton s n the soluton of sets of smultaneous lnear ordnary dfferental equatons, for example, equatons lke () () () my + cy + ky = f t ρv + αy = g t can be put n the form x = Ax+ b t Lq + Cv + By = h t where A s a properly defned square matrx, and xb, are properly defned column vectors. ()

18 Smlarty Transformatons Sometmes t s useful to solve the equatons n a coordnate system that s dfferent from the orgnal problem formulaton. Any square nonsngular matrx T can be consdered a transformaton matrx. 1 x Tx, x T x () 1 1 () Lnear coordnate transformatons of column vectors are accomplshed va transformatons = = For example, under such a transformaton x = Ax+ b t x = T ATx + T b t Matrces transform under a change of 1 A T AT = Ths s called a smlarty transformaton. coordnates accordng to

19 Specal Matrces Smlarty transformatons are used to transform matrces nto a varety of specal forms (when possble). Among these are: a11 a 0 ann lower companon: an1 an2 a nn 0 22 dagonal:, upper trangular: a a a 0 a a a n 22 2n nn

20 Advanced Topcs ~ Defer Egenvalues/Egenvectors Functons of Matrces Cayley-Hamlton Theorem Sngular Values

21 Summary Vectors Basc defntons & operatons lnear dependent\ndependent sets Matrces Defntons Algebrac operatons Determnants, Rank, Inverse Smlarty transformatons & specal matrc forms MATLAB functons

### 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,

### 8.4 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

### 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

### 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

### 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,

### 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

### 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

### = = = (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

### Norms, 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

### 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

### 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

### 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

### 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

### Section 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

### 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....................................

### 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

### 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

### Linear 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

### 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

### 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

### 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.

### Bezier 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

### 763622S 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 Â are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)

### 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

### The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices

Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan

### 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

### Number of cases Number of factors Number of covariates Number of levels of factor i. Value of the dependent variable for case k

ANOVA Model and Matrx Computatons Notaton The followng notaton s used throughout ths chapter unless otherwse stated: N F CN Y Z j w W Number of cases Number of factors Number of covarates Number of levels

### 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

### Fall 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

### 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

### 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

### 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

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

### Time-Varying Systems and Computations Lecture 6

Tme-Varyng 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

### Bézier 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

### Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

### Singular 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

### SUMMARY OF STOICHIOMETRIC RELATIONS AND MEASURE OF REACTIONS' PROGRESS AND COMPOSITION FOR MULTIPLE REACTIONS

UMMAY OF TOICHIOMETIC ELATION AND MEAUE OF EACTION' POGE AND COMPOITION FO MULTIPLE EACTION UPDATED 0/4/03 - AW APPENDIX A. In case of multple reactons t s mportant to fnd the number of ndependent reactons.

### 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

### 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

### MEM633 Lectures 7&8. Chapter 4. Descriptions of MIMO Systems 4-1 Direct Realizations. (i) x u. y x

MEM633 Lectures 7&8 Chapter 4 Descrptons of MIMO Systems 4- Drect ealzatons y() s s su() s y () s u () s ( s)( s) s y() s u (), s y() s u() s s s y() s u(), s y() s u() s ( s)( s) s () ( s ) y ( s) u (

### 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

### Integrals and Invariants of Euler-Lagrange Equations

Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,

### 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

### 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

### THE 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

### The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method

Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse

### 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

### 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

### 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

### Section 3.6 Complex Zeros

04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

### Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented

### Workshop: 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

### 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

### (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 steppng-o pont for understandng neural networks. Ths handout ncludes basc dentons, then quckly progresses to elementary

### 1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse

### Randić Energy and Randić Estrada Index of a Graph

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL

### 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

### Unit 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

### THEOREMS OF QUANTUM MECHANICS

THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn

### 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

### 12 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

### Algebraic Solutions to Multidimensional Minimax Location Problems with Chebyshev Distance

Recent Researches n Appled and Computatonal Mathematcs Algebrac Solutons to Multdmensonal Mnmax Locaton Problems wth Chebyshev Dstance NIKOLAI KRIVULIN Faculty of Mathematcs and Mechancs St Petersburg

### 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

### 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

### 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

### Lecture 3: Dual problems and Kernels

Lecture 3: Dual problems and Kernels C4B Machne Learnng Hlary 211 A. Zsserman Prmal and dual forms Lnear separablty revsted Feature mappng Kernels for SVMs Kernel trck requrements radal bass functons SVM

### 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

### A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b

Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 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,

### CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13

CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,

### 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

### Lecture 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

### MMA and GCMMA two methods for nonlinear optimization

MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons

### 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

### Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q

For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals

### MATH 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

### THE 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

### 8.6 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

### CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

### 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

### 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

### Lecture 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

### Fundamental loop-current method using virtual voltage sources technique for special cases

Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,

### 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/

### LINEAR 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

### Statistical Mechanics and Combinatorics : Lecture III

Statstcal Mechancs and Combnatorcs : Lecture III Dmer Model Dmer defntons Defnton A dmer coverng (perfect matchng) of a fnte graph s a set of edges whch covers every vertex exactly once, e every vertex

### 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

### 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,

### Outline and Reading. Dynamic Programming. Dynamic Programming revealed. Computing Fibonacci. The General Dynamic Programming Technique

Outlne and Readng Dynamc Programmng The General Technque ( 5.3.2) -1 Knapsac Problem ( 5.3.3) Matrx Chan-Product ( 5.3.1) Dynamc Programmng verson 1.4 1 Dynamc Programmng verson 1.4 2 Dynamc Programmng

### A new Approach for Solving Linear Ordinary Differential Equations

, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of

### Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )

Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often

### 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

### 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

### Some Consequences. Example of Extended Euclidean Algorithm. The Fundamental Theorem of Arithmetic, II. Characterizing the GCD and LCM

Example of Extended Eucldean Algorthm Recall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to wrte 3 as a lnear combnaton of 84 and 33: 3 = 18 15 [Now 3 s

### Difference 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

### Deriving the X-Z Identity from Auxiliary Space Method

Dervng the X-Z 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

### 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

### 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