Contents. Outline. Structured Rank Matrices Lecture 2: The theorem Proofs Examples related to structured ranks References. Structure Transport


 Vanessa Montgomery
 3 years ago
 Views:
Transcription
1 Contents Structured Rnk Mtrices Lecture 2: Mrc Vn Brel nd Rf Vndebril Dept. of Computer Science, K.U.Leuven, Belgium Chemnitz, Germny, September Exmples relted to structured rnks 2 2 / 26 Outline 1 Exmples relted to structured rnks 2 Definition (Right null spce) Given mtrix A R m n. The right null spce N(A) equls Definition (Nullity of mtrix) N(A) = {x R n Ax = 0}. Given mtrix A R m n. The nullity n(a) is defined s the dimension of the right null spce of A. Corollry The dimension of the right null spce corresponds to the rnk deficiency of the columns of the mtrix A: n(a) = n rnk (A) = (number of columns) rnk (A). 3 / 26 4 / 26
2 Theorem (Nullity theorem) Suppose the following invertible mtrix A R n n is prtitioned s [ ] A11 A A = 12 A 21 A 22 with A 11 of size p q. The inverse B of A is prtitioned s [ ] A 1 B11 B = B = 12 B 21 B 22 with B 11 of size q p. Then the nullities n(a 11 ) nd n(b 22 ) re equl: n(a 11 ) = n(b 22 ). Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Proof: Permuting the mtrix such tht A(N\β; ) moves to the upper left position A 11, will move A 1 (α; β) to the position B 22. Using the equlities: gives us the proof. n(a 11 ) = n α rnk (A 11 ), n(b 22 ) = β rnk (B 22 ), 4 / 26 5 / 26 Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Exmples for 5 5 mtrices: α = {1, 2} nd N\β = {3, 4, 5} nd β = {1, 2} = {3, 4, 5} Exmples for 5 5 mtrices: α = {1, 2} nd N\β = {4, 5} nd β = {1, 2, 3} = {3, 4, 5} 5 / 26 5 / 26
3 Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Corollry Corollries of the nullity theorem Suppose A R n n is nonsingulr mtrix, nd α, β re nonempty subsets of N with α < n nd β < n. Then rnk ( A 1 (α; β) ) = rnk (A(N\β; )) + α + β n. Exmples for 5 5 mtrices: α = {3, 4, 5} nd N\β = {3, 4, 5} nd β = {1, 2} = {1, 2} Exmples for 5 5 mtrices: α = {2, 4} nd N\β = {2, 4, 5} nd β = {1, 3} = {1, 3, 5} 5 / 26 5 / 26 Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Proof: Is direct consequence of the previous eqution: ( ) rnk A 1 (α; β) = rnk (A(N\β; )) + α + β n, when posing β = : ( ) rnk A 1 (α; ) = rnk (A(α; )) + α + n. This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {2, 3, 4, 5} nd α = {3, 4, 5} nd = {1} = {1, 2} 6 / 26 6 / 26
4 Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). Some corollries of the nullity theorem Corollry For nonsingulr mtrix A R n n nd α N, we hve: rnk ( A 1 (α; ) ) = rnk (A(α; )). This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {4, 5} nd α = {5} nd = {1, 2, 3} = {1, 2, 3, 4} This mens tht for mtrix the following blocks lwys hve the sme rnk in A nd in A 1. α = {3, 5} nd α = {2, 3} nd = {1, 2, 4} = {1, 4, 5} 6 / 26 6 / 26 Outline 1 Exmples relted to structured rnks 2 Different proofs There exist different strtegies to prove the nullity theorem. An importnt remrk, the theorem predicts structures but does not provide inversion formuls. Fiedler nd Mrkhm proved it, working directly on the rnks nd nullities of the blocks, their proof ws bsed on pper by Gustfson. Brrett nd Feinsilver were very close to n lterntive proof, but they only worked with tridigonl nd semiseprble mtrices. Recently lso Strng nd Nguyen proved weker formultion of the theorem. 7 / 26 8 / 26
5 Different proofs Proof (by Fiedler nd Mrkhm) Suppose n(a 11 ) n(b 22 ). If this is not true, we cn prove the theorem for the mtrices [ ] [ ] A22 A 21 B22 B, 21, A 12 A 11 B 12 B 11 which re lso ech others inverse. Suppose n(b 22 ) > 0 otherwise n(a 11 ) = 0 nd the theorem is proved. When n(b 22 ) = c > 0, then there exists mtrix F with c linerly independent columns, such tht B 22 F = 0. Remember tht [ ] [ ] [ ] A11 A 12 B11 B 12 I 0 =. A 21 A 22 B 21 B 22 0 I Different proofs Proof (by Fiedler nd Mrkhm) Hence, multiplying the following eqution to the right by F we get A 11 B 12 + A 12 B 22 = 0, Applying the sme opertion to the reltion: A 11 B 12 F = 0. (1) A 21 B 12 + A 22 B 22 = I it follows tht A 21 B 12 F = F, nd therefore rnk (B 12 F ) c. Using this lst sttement together with eqution (1), we derive n(a 11 ) rnk (B 12 F ) c = n(b 22 ). 9 / 26 With our ssumption n(a 11 ) n(b 22 ), this proves the theorem. 9 / 26 Outline 1 Exmples relted to structured rnks Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix / / 26
6 Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26 Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry Exmple (Upper tringulr mtrix) The inverse of n upper tringulr mtrix is n upper tringulr mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26
7 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry 2. Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry / / 26
8 Exmple (Qusiseprble mtrix) The inverse of qusiseprble mtrix is qusiseprble mtrix. The rnk of the red mrked blocks is mintined by Corollry 2. Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. 12 / / 26 Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {3, 4, 5} nd N\β = {2, 3, 4, 5} nd β = {1} = {1, 2} Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {4, 5} nd N\β = {3, 4, 5} nd β = {1, 2} = {1, 2, 3} / / 26
9 Exmple (Tridigonl vs. semiseprble) The inverse of tridigonl mtrix is semiseprble mtrix. The rnk of the left block plus 1 equls the rnk of the right block, ccording to corollry 1 α = {5} nd N\β = {4, 5} nd β = {1, 2, 3} = {1, 2, 3, 4} Exmple The inverse of {p, q}semiseprble mtrix is {p, q}bnd mtrix. One cn predict the structure of the inverse of generlized Hessenberg mtrix. One cn predict the structure when inverting hierrchiclly semiseprble nd/or H mtrices. Structure relted: The offdigonl structure is mintined. For exmple the inverse of rnk one mtrix plus digonl is gin rnk 1 mtrix plus digonl. Applicble to ll structured rnk mtrices. 13 / / 26 Outline for the nullity theorem 1 Exmples relted to structured rnks 2 W. H. Gustfson, A note on mtrix inversion, Liner Algebr nd Its Applictions 57 (1984), M. Fiedler, Bsic mtrices, Liner Algebr nd Its Applictions 373 (2003), W. W. Brrett, A theorem on inverse of tridigonl mtrices, Liner Algebr nd Its Applictions 27 (1979), W. W. Brrett nd P. J. Feinsilver, Gussin fmilies nd theorem on ptterned mtrices, Journl of Applied Probbility 15 (1978), W. W. Brrett nd P. J. Feinsilver, Inverses of bnded mtrices, Liner Algebr nd Its Applictions 41 (1981), G. Strng nd T. Nguyen, The interply of rnks of submtrices, SIAM Review 46 (2004), / / 26
10 Generl remrks LU nd QRdecompositions Given mtrix A R m n. A = LU is clled n LUdecomposition if L is lower tringulr nd U is upper tringulr. Frequently used for solving systems of equtions (Gussin elimintion). Computing eigenvlues of specilized mtrices (quotientdifference lgorithms). Generl remrks LU nd QRdecompositions Given mtrix A R m n. A = LU is clled n LUdecomposition if L is lower tringulr nd U is upper tringulr. Frequently used for solving systems of equtions (Gussin elimintion). Computing eigenvlues of specilized mtrices (quotientdifference lgorithms). Given mtrix A R m n. A = QR is clled QRdecomposition if Q is unitry (QQ H = Q H Q = I ) nd R is upper tringulr. Solving systems of equtions (more stble thn Gussin eliminition). In the top 10 lgorithms of the 20th century for computing eigenvlues of rbitrry mtrices. Under some mild conditions both fctoriztions re unique. Under some mild conditions both fctoriztions re unique. 17 / / 26 Outline 1 Exmples relted to structured rnks 2 Theorem (LUfctoriztion) Given n invertible mtrix A, with LU fctoriztion A = LU. Let A be prtitioned s [ ] A11 A A = 12 A 21 A 22 with A 11 of dimension p q. Let U be prtitioned s [ ] U11 U U = 12 0 U 22 with U 11 of dimension p q. Then the nullities n(a 12 ) nd n(u 12 ) re equl (s well s their rnks). 18 / / 26
11 Outline Exmple (Structured rnk mtrices) The L nd U fctor inherit the structure. For semiseprble mtrix: U is upper semiseprble, nd L is lower semiseprble. For tridigonl mtrix: U is upper bidigonl, nd L is lower bidigonl. For {p, q}semiseprble mtrix: U is {q}upper semiseprble, nd L is {p}lower semiseprble. For {p, q}bnd mtrix: U is {q}upper bnd, nd L is {p}lower bnd. Holds for combintions, nd even more generl structures. 1 Exmples relted to structured rnks 2 20 / / 26 Theorem (QRfctoriztion) Given n invertible mtrix A, with QRfctoriztion A = QR. Let A be prtitioned s [ ] A11 A A = 12, A 21 A 22 with A 11 of dimension p q. Let Q be prtitioned s [ ] Q11 Q Q = 12, Q 21 Q 22 with Q 11 of dimension p q. Then the nullities n(a 21 ) nd n(q 21 ) re equl. Exmple (Structured rnk mtrices) The Q fctor inherits the structure of the lower tringulr prt. The structure of R is more complicted (see next slides). For semiseprble mtrix: Q hs the lower tringulr prt of lower semiseprble form, nd R hs the upper tringulr structure of rnk 2. For tridigonl mtrix: Q hs the lower tringulr prt of bidigonl form. For {p, q}semiseprble mtrix: Q hs the lower tringulr prt of {p}semiseprble form. For {p, q}bnd mtrix: Q hs the lower tringulr prt of {p}bnd form. Holds for combintions, nd even more generl structures. 22 / / 26
12 Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Strting sitution: Rk r Rk s Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. First series of Givens trnsformtions: Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. First series of Givens trnsformtions: Q 1 Rk r Rk s Rk s = Rk r Q 1 Rk s
13 Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. First series of Givens trnsformtions: Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Second series of Givens trnsformtions: Q 2 Rk s = Rk r Q 1 Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Second series of Givens trnsformtions: Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Third series of Givens trnsformtions: Q 2 = Q 3
14 Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Third series of Givens trnsformtions: Rnk structure of the Rfctor We derive this structure by investigting how the originl rnk structure is trnsformed when computing the QRfctoriztion. Third series of Givens trnsformtions: = Q 3 = Q 3 Q.E.D. Outline 1 Exmples relted to structured rnks for these generliztions R. Vndebril nd M. Vn Brel, A short note on the nullity theorem, Journl of Computtionl nd Applied Mthemtics 189: , / / 26
Elements of Matrix Algebra
Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 201516 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
More informationMath Lecture 23
Mth 8  Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationEngineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: TuTh 11:0012:00
Engineering Anlysis ENG 3420 Fll 2009 Dn C. Mrinescu Office: HEC 439 B Office hours: TuTh 11:0012:00 Lecture 13 Lst time: Problem solving in preprtion for the quiz Liner Algebr Concepts Vector Spces,
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationChapter 5 Determinants
hpter 5 Determinnts 5. Introduction Every squre mtri hs ssocited with it sclr clled its determinnt. Given mtri, we use det() or to designte its determinnt. We cn lso designte the determinnt of mtri by
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More information308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:
8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationarxiv: v2 [math.nt] 2 Feb 2015
rxiv:407666v [mthnt] Fe 05 Integer Powers of Complex Tridigonl AntiTridigonl Mtrices Htice Kür Duru &Durmuş Bozkurt Deprtment of Mthemtics, Science Fculty of Selçuk University Jnury, 08 Astrct In this
More informationThe Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11
The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd GussSiedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic
More informationIntroduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices
Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationLecture Note 9: Orthogonal Reduction
MATH : Computtionl Methods of Liner Algebr 1 The Row Echelon Form Lecture Note 9: Orthogonl Reduction Our trget is to solve the norml eution: Xinyi Zeng Deprtment of Mthemticl Sciences, UTEP A t Ax = A
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationSemigroup of generalized inverses of matrices
Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More informationAlgebra Of Matrices & Determinants
lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix  bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in
More informationMTH 5102 Linear Algebra Practice Exam 1  Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm  Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
More informationPart IB Numerical Analysis
Prt IB Numericl Anlysis Theorems with proof Bsed on lectures by G. Moore Notes tken by Dexter Chu Lent 2016 These notes re not endorsed by the lecturers, nd I hve modified them (often significntly) fter
More informationMultivariate problems and matrix algebra
University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationComputing The Determinants By Reducing The Orders By Four
Applied Mthemtics ENotes, 10(2010), 151158 c ISSN 16072510 Avilble free t mirror sites of http://wwwmthnthuedutw/ men/ Computing The Determinnts By Reducing The Orders By Four Qefsere Gjonblj, Armend
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationOn the free product of ordered groups
rxiv:703.0578v [mth.gr] 6 Mr 207 On the free product of ordered groups A. A. Vinogrdov One of the fundmentl questions of the theory of ordered groups is wht bstrct groups re orderble. E. P. Shimbirev [2]
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationThe Algebra (aljabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (ljbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationLinear Algebra 1A  solutions of ex.4
Liner Algebr A  solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists  ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationResults on Planar Near Rings
Interntionl Mthemticl Forum, Vol. 9, 2014, no. 23, 11391147 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/imf.2014.4593 Results on Plnr Ner Rings Edurd Domi Deprtment of Mthemtics, University
More informationA Criterion on Existence and Uniqueness of Behavior in Electric Circuit
Institute Institute of of Advnced Advnced Engineering Engineering nd nd Science Science Interntionl Journl of Electricl nd Computer Engineering (IJECE) Vol 6, No 4, August 2016, pp 1529 1533 ISSN: 20888708,
More informationNumerical Methods I Orthogonal Polynomials
Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATHGA 2011.003 / CSCIGA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationPath product and inverse Mmatrices
Electronic Journl of Liner Algebr Volume 22 Volume 22 (2011) Article 42 2011 Pth product nd inverse Mmtrices Yn Zhu ChengYi Zhng Jun Liu Follow this nd dditionl works t: http://repository.uwyo.edu/el
More informationLINEAR ALGEBRA AND MATRICES. n ij. is called the main diagonal or principal diagonal of A. A column vector is a matrix that has only one column.
PART 1 LINEAR ALGEBRA AND MATRICES Generl Nottions Mtri (denoted by cpitl boldfce letter) A is n m n mtri. 11 1... 1 n 1... n A ij...... m1 m... mn ij denotes the component t row i nd column j of A. If
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More informationModified Crout s method for an LU decomposition of an interval matrix
Journl of Physics: Conference Series PAPER OPEN ACCESS Modified Crout s method for n decomposition of n intervl mtrix To cite this rticle: T Nirml nd K Gnesn 2018 J. Phys.: Conf. Ser. 1000 012134 View
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationStability of block LDL T factorization of a symmetric tridiagonal matrix
Liner Algebr nd its Applictions 287 (1999) 181±189 Stbility of block LDL T fctoriztion of symmetric tridigonl mtrix Nichols J. Highm 1 Deprtment of Mthemtics, University of Mnchester, Mnchester M13 9PL,
More informationCHAPTER 1 PROGRAM OF MATRICES
CHPTER PROGRM OF MTRICES  INTRODUCTION definition of engineering is the science y which the properties of mtter nd sources of energy in nture re mde useful to mn. Thus n engineer will hve to study the
More informationMAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationDonnishJournals
DoishJournls 20411189 Doish Journl of Eductionl Reserch nd Reviews Vol 2(1) pp 001007 Jnury, 2015 http://wwwdoishjournlsorg/djerr Copyright 2015 Doish Journls Originl Reserch Article Algebr of Mtrices
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2
CS434/54: Pttern Recognition Prof. Olg Veksler Lecture Outline Review of Liner Algebr vectors nd mtrices products nd norms vector spces nd liner trnsformtions eigenvlues nd eigenvectors Introduction to
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationMatrices 13: determinant properties and rules continued
Mtrices : determinnt properties nd rules continued nthony Rossiter http://controleduction.group.shef.c.uk/indexwebbook.html http://www.shef.c.uk/cse Deprtment of utomtic Control nd Systems Engineering
More informationA new algorithm for generating Pythagorean triples 1
A new lgorithm for generting Pythgoren triples 1 RH Dye 2 nd RWD Nicklls 3 The Mthemticl Gzette (1998; 82 (Mrch, No. 493, pp. 86 91 http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf 1 Introduction
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More information6.2 The Pythagorean Theorems
PythgorenTheorems20052006.nb 1 6.2 The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this
More information4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.
4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)
More informationGeneralized Fano and nonfano networks
Generlized Fno nd nonfno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More informationLesson 55  Inverse of Matrices & Determinants
// () Review Lesson  nverse of Mtries & Determinnts Mth Honors  Sntowski  t this stge of stuying mtries, we know how to, subtrt n multiply mtries i.e. if Then evlute: () + B (b)  () B () B (e) B n
More informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the evectors nd evlues
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationUNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More information1 Orthogonalisation in finite precision arithmetic
1 Orthogonlistion in finite precision rithmetic We investigte the differences nd similrities between the following four wys to compute the QRdecomposition of given rectngulr mtrix A C m n in Mtlb: (CGS)
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More information