Miscellaneous Results, Solving Equations, and Generalized Inverses. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
|
|
- Dustin Booth
- 5 years ago
- Views:
Transcription
1 Miscellaneous Results, Solving Equations, and Generalized Inverses opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
2 Result A.7: Suppose S and T are vector spaces. If S T and dim(s) = dim(t ), then S = T. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
3 Result A.8: Suppose m n A and b satisfy m 1 Then m n A = m n 0 and b = 0. m 1 m 1 Ax + b = 0 x R n. m n m 1 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
4 Corollary A.1: If m n B and m n C satisfy Bx = Cx x R n, then B = C. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
5 Corollary A.2: Suppose m n A has full column rank. Then AB = AC = B = C. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
6 Lemma A.1: C C = 0 = C = 0. Proof of Lemma A.1: Let c i denote the i th column of C. Then the i th diagonal element of C C is c i c i. C C = 0, c i c i = 0 i = 1,..., n. Now c i c i = 0 i = c i = 0 i = C = 0. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
7 Another Result on the Rank of a Product Suppose rank( m n A ) = n and rank( C ) = k. Then k l rank( B ) = rank(abc). n k Proof: HW problem. Corollaries: If A is full-column rank, rank(ab) = rank(b). If C is full-row rank, rank(bc) = rank(b). opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
8 Solving Equations: Consider a system of linear equations Ax = c, where m n A is a known matrix and c is a known vector. m 1 opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
9 We seek a solution vector x that satisfies Ax = c. If m = n so that m n A is a square and if m n A is nonsingular, then A 1 Ax = A 1 c = x = A 1 c is the unique solution to Ax = c. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
10 If m n A is singular or not square, Ax = c may have no solution or infinitely many solutions or a unique solution. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
11 A system of equations Ax = c is consistent if there exists a solution x such that Ax = c. A systems of equation Ax = c is inconsistent if Ax c x R n. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
12 Result A.9: A system of equations Ax = c is consistent iff c C(A). Provide an example m n A, c Ax = c is inconsistent. m 1 Provide an example m n A, c Ax = c is consistent. m 1 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
13 Generalized Inverse A matrix G is a generalized inverse (GI) of a matrix A iff AGA = A. Every matrix has at least one GI. We will use A to denote a GI of a matrix A. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
14 If A is nonsingular, then A 1 is the unique GI of A. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
15 Result A.10: Suppose rank( m n A) = r. If A can be partitioned as A = where rank( r r C) = r, then C r r E (m r) r G = n m D r (n r) F (m r) (n r) [ ] C , is a GI of m n A. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
16 Proof of Result A.10: [ ] [ ] C D C 1 0 AG = E F 0 0 [ ] I 0 =. EC 1 0 [ ] [ ] [ ] I 0 C D C D AGA = =. EC 1 0 E F E EC 1 D We need to show EC 1 D = F. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
17 First note that rank( r r C) = r = rank([c, D]) = r [C, D] has at least r LI columns and at most r LI rows. (rank([c, D]) r and rank([c, D]) r = rank([c, D]) = r.) Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
18 ([ ]) C D Now rank = r = each row of [E, F] is a LC of the rows of E F [C, D]. Thus a matrix K K[C, D] = [E, F] [KC, KD] = [E, F] KC = E, KD = F. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
19 Now KC = E K = EC 1. Together with KD = F, this implies EC 1 D = F. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
20 Permutation Matrix A matrix n n P is a permutation matrix if the rows of P are the same as the rows of n n I but not necessarily in the same order. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
21 Example: P = If A = , then PA = Order of rows of PA are permuted relative to order of rows of A. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
22 Example: [ ] P = , B = [ ] Then BP = Order of columns of BP are permuted relative to order of columns of B. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
23 A Permutation Matrix is Nonsingular The rows (and columns) of a permutation matrix are the same as those of the identity matrix. Thus, a permutation matrix has full rank and is therefore nonsingular. Furthermore, if P = [p 1,..., p n ] is a permutation matrix, P 1 = P p 0 if i j ip j = 1 if i = j. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
24 Result A.11: Suppose rank( m n A) = r. There exist permutation matrices m m P and Q such n n that [ ] C D PAQ =, E F where rank( r r C) = r. Furthermore, is a GI of A. [ ] C 1 0 G = Q P 0 0 opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
25 Proof of Result A.11: Because rank(a) = r, there exists a set of r rows of A that are LI. Let P be a permutation matrix, the first r rows of PA are LI. Let H be the matrix consisting of the first r rows of PA. Then rank(h) = r. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
26 This implies that a set of r columns of H that are LI. Let Q be a permutation matrix the first r columns of HQ are LI. Then the submatrix consisting of the first r rows and first r columns of PAQ has rank r. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
27 [ ] C D Thus we can partition PAQ as, where rank( E F r r C) = r. [ ] [ ] C 1 0 C D By Result A.10, is a GI for PAQ =. 0 0 E F Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
28 [ ] [ ] C D C D PAQ = A = P 1 Q 1 E F E F [ ] C 1 0 AQ PA = 0 0 [ ] [ ] [ C D C = P 1 Q C Q PP 1 E F 0 0 E [ ] [ ] [ ] C D C = P C D Q 1 E F 0 0 E F [ ] C D = P 1 Q 1 = A. E F ] D F Q 1 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
29 Use Result A.11 to find a GI for A = Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
30 Algorithm for finding a GI of A: 1. Find an r r nonsingular submatrix of A, where r = rank(a). Call this matrix W. 2. Compute (W 1 ). 3. Replace each element of W in A with the corresponding elements of (W 1 ). 4. Replace all other elements in A with zeros. 5. Transpose resulting matrix to get A. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
31 Result A.12: Let Ax = c be a consistent system of equations, and let G be any GI of A. Then Gc is a solution to Ax = c, i.e., AGc = c. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
32 Result A.13: Let Ax = c be a consistent system of equations, and let G be any GI of A. Then x is a solution to Ax = c iff z x = Gc + (I GA)z. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
33 Prove that a consistent system of equations Ax = c has a unique solution if and only if A is full-column rank and infinitely many solutions if and only if A is less than full-column rank. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
Constraints on Solutions to the Normal Equations. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 41
Constraints on Solutions to the Normal Equations Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 41 If rank( n p X) = r < p, there are infinitely many solutions to the NE X Xb
More informationPreliminary Linear Algebra 1. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 100
Preliminary Linear Algebra 1 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 100 Notation for all there exists such that therefore because end of proof (QED) Copyright c 2012
More informationGeneral Linear Test of a General Linear Hypothesis. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 35
General Linear Test of a General Linear Hypothesis Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 35 Suppose the NTGMM holds so that y = Xβ + ε, where ε N(0, σ 2 I). opyright
More informationPreliminaries. Copyright c 2018 Dan Nettleton (Iowa State University) Statistics / 38
Preliminaries Copyright c 2018 Dan Nettleton (Iowa State University) Statistics 510 1 / 38 Notation for Scalars, Vectors, and Matrices Lowercase letters = scalars: x, c, σ. Boldface, lowercase letters
More informationDeterminants. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 25
Determinants opyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 25 Notation The determinant of a square matrix n n A is denoted det(a) or A. opyright c 2012 Dan Nettleton (Iowa State
More informationEstimation of the Response Mean. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 27
Estimation of the Response Mean Copyright c 202 Dan Nettleton (Iowa State University) Statistics 5 / 27 The Gauss-Markov Linear Model y = Xβ + ɛ y is an n random vector of responses. X is an n p matrix
More informationChapter 1. Matrix Algebra
ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface
More informationSection 3.3. Matrix Rank and the Inverse of a Full Rank Matrix
3.3. Matrix Rank and the Inverse of a Full Rank Matrix 1 Section 3.3. Matrix Rank and the Inverse of a Full Rank Matrix Note. The lengthy section (21 pages in the text) gives a thorough study of the rank
More informationDistributions of Quadratic Forms. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 31
Distributions of Quadratic Forms Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 31 Under the Normal Theory GMM (NTGMM), y = Xβ + ε, where ε N(0, σ 2 I). By Result 5.3, the NTGMM
More informationEstimable Functions and Their Least Squares Estimators. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 51
Estimable Functions and Their Least Squares Estimators Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 51 Consider the GLM y = n p X β + ε, where E(ε) = 0. p 1 n 1 n 1 Suppose
More informationML and REML Variance Component Estimation. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 58
ML and REML Variance Component Estimation Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 58 Suppose y = Xβ + ε, where ε N(0, Σ) for some positive definite, symmetric matrix Σ.
More informationThe Multivariate Normal Distribution. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 36
The Multivariate Normal Distribution Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 36 The Moment Generating Function (MGF) of a random vector X is given by M X (t) = E(e t X
More informationThe Multivariate Normal Distribution. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 36
The Multivariate Normal Distribution Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 36 The Moment Generating Function (MGF) of a random vector X is given by M X (t) = E(e t X
More informationScheffé s Method. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37
Scheffé s Method opyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 37 Scheffé s Method: Suppose where Let y = Xβ + ε, ε N(0, σ 2 I). c 1β,..., c qβ be q estimable functions, where
More informationApplied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University
Applied Matrix Algebra Lecture Notes Section 22 Gerald Höhn Department of Mathematics, Kansas State University September, 216 Chapter 2 Matrices 22 Inverses Let (S) a 11 x 1 + a 12 x 2 + +a 1n x n = b
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationThe matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.
) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to
More informationLikelihood Ratio Test of a General Linear Hypothesis. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 42
Likelihood Ratio Test of a General Linear Hypothesis Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 42 Consider the Likelihood Ratio Test of H 0 : Cβ = d vs H A : Cβ d. Suppose
More informationLinear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary
More informationMATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:
More informationXβ is a linear combination of the columns of X: Copyright c 2010 Dan Nettleton (Iowa State University) Statistics / 25 X =
The Gauss-Markov Linear Model y Xβ + ɛ y is an n random vector of responses X is an n p matrix of constants with columns corresponding to explanatory variables X is sometimes referred to as the design
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition. 2 Inverse Existence and Uniqueness
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationMatrix Algebra, part 2
Matrix Algebra, part 2 Ming-Ching Luoh 2005.9.12 1 / 38 Diagonalization and Spectral Decomposition of a Matrix Optimization 2 / 38 Diagonalization and Spectral Decomposition of a Matrix Also called Eigenvalues
More informationANOVA Variance Component Estimation. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 32
ANOVA Variance Component Estimation Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 32 We now consider the ANOVA approach to variance component estimation. The ANOVA approach
More informationEp Matrices and Its Weighted Generalized Inverse
Vol.2, Issue.5, Sep-Oct. 2012 pp-3850-3856 ISSN: 2249-6645 ABSTRACT: If A is a con s S.Krishnamoorthy 1 and B.K.N.MuthugobaI 2 Research Scholar Ramanujan Research Centre, Department of Mathematics, Govt.
More informationMath 102 Final Exam - Dec 14 - PCYNH pm Fall Name Student No. Section A0
Math 12 Final Exam - Dec 14 - PCYNH 122-6pm Fall 212 Name Student No. Section A No aids allowed. Answer all questions on test paper. Total Marks: 4 8 questions (plus a 9th bonus question), 5 points per
More informationANOVA Variance Component Estimation. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 32
ANOVA Variance Component Estimation Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 32 We now consider the ANOVA approach to variance component estimation. The ANOVA approach
More informationSTAT200C: Review of Linear Algebra
Stat200C Instructor: Zhaoxia Yu STAT200C: Review of Linear Algebra 1 Review of Linear Algebra 1.1 Vector Spaces, Rank, Trace, and Linear Equations 1.1.1 Rank and Vector Spaces Definition A vector whose
More informationELE/MCE 503 Linear Algebra Facts Fall 2018
ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationOn the necessary and sufficient condition for the extended Wedderburn-Guttman theorem
On the necessary and sufficient condition for the extended Wedderburn-Guttman theorem Yoshio Takane a,1, Haruo Yanai b a Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal,
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationMTH Midterm 2 Review Questions - Solutions
MTH 235 - Midterm 2 Review Questions - Solutions The following questions will be good practice for some of the questions you will see on the midterm. However, doing these problems alone is not nearly enough.
More informationACI-matrices all of whose completions have the same rank
ACI-matrices all of whose completions have the same rank Zejun Huang, Xingzhi Zhan Department of Mathematics East China Normal University Shanghai 200241, China Abstract We characterize the ACI-matrices
More informationPaul Schrimpf. September 10, 2013
Systems of UBC Economics 526 September 10, 2013 More cardinality Let A and B be sets, the Cartesion product of A and B is A B := {(a, b) : a A, b B} Question If A and B are countable is A B countable?
More informationElementary operation matrices: row addition
Elementary operation matrices: row addition For t a, let A (n,t,a) be the n n matrix such that { A (n,t,a) 1 if r = c, or if r = t and c = a r,c = 0 otherwise A (n,t,a) = I + e t e T a Example: A (5,2,4)
More informationMATH 532: Linear Algebra
MATH 532: Linear Algebra Chapter 4: Vector Spaces Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2015 fasshauer@iit.edu MATH 532 1 Outline 1 Spaces and Subspaces
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
More informationMath 308 Discussion Problems #4 Chapter 4 (after 4.3)
Math 38 Discussion Problems #4 Chapter 4 (after 4.3) () (after 4.) Let S be a plane in R 3 passing through the origin, so that S is a two-dimensional subspace of R 3. Say that a linear transformation T
More informationSolutions to Exam I MATH 304, section 6
Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More informationThe Gauss-Markov Model. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 61
The Gauss-Markov Model Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 61 Recall that Cov(u, v) = E((u E(u))(v E(v))) = E(uv) E(u)E(v) Var(u) = Cov(u, u) = E(u E(u)) 2 = E(u 2
More informationThe Aitken Model. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 41
The Aitken Model Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 41 The Aitken Model (AM): Suppose where y = Xβ + ε, E(ε) = 0 and Var(ε) = σ 2 V for some σ 2 > 0 and some known
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More informationLinear Algebra Formulas. Ben Lee
Linear Algebra Formulas Ben Lee January 27, 2016 Definitions and Terms Diagonal: Diagonal of matrix A is a collection of entries A ij where i = j. Diagonal Matrix: A matrix (usually square), where entries
More informationDiscrete Optimization 23
Discrete Optimization 23 2 Total Unimodularity (TU) and Its Applications In this section we will discuss the total unimodularity theory and its applications to flows in networks. 2.1 Total Unimodularity:
More informationEstimating Estimable Functions of β. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 17
Estimating Estimable Functions of β Copyright c 202 Dan Nettleton (Iowa State University) Statistics 5 / 7 The Response Depends on β Only through Xβ In the Gauss-Markov or Normal Theory Gauss-Markov Linear
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationMath 407: Linear Optimization
Math 407: Linear Optimization Lecture 16: The Linear Least Squares Problem II Math Dept, University of Washington February 28, 2018 Lecture 16: The Linear Least Squares Problem II (Math Dept, University
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationMathematical Foundations of Applied Statistics: Matrix Algebra
Mathematical Foundations of Applied Statistics: Matrix Algebra Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/105 Literature Seber, G.
More informationCHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then
LECTURE 6: POSITIVE DEFINITE MATRICES Definition: A Hermitian matrix A Mn is positive definite (pd) if x Ax > 0 x C n,x 0 A is positive semidefinite (psd) if x Ax 0. Definition: A Mn is negative (semi)definite
More informationMATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS
MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on
More informationNotes on Row Reduction
Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University September The Row-Reduction Algorithm The row-reduced form of a matrix contains a great deal of information, both
More informationMTH 102A - Linear Algebra II Semester
MTH 0A - Linear Algebra - 05-6-II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )
More informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More information(1) for all (2) for all and all
8. Linear mappings and matrices A mapping f from IR n to IR m is called linear if it fulfills the following two properties: (1) for all (2) for all and all Mappings of this sort appear frequently in the
More informationEE731 Lecture Notes: Matrix Computations for Signal Processing
EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationGATE Engineering Mathematics SAMPLE STUDY MATERIAL. Postal Correspondence Course GATE. Engineering. Mathematics GATE ENGINEERING MATHEMATICS
SAMPLE STUDY MATERIAL Postal Correspondence Course GATE Engineering Mathematics GATE ENGINEERING MATHEMATICS ENGINEERING MATHEMATICS GATE Syllabus CIVIL ENGINEERING CE CHEMICAL ENGINEERING CH MECHANICAL
More informationThe skew-symmetric orthogonal solutions of the matrix equation AX = B
Linear Algebra and its Applications 402 (2005) 303 318 www.elsevier.com/locate/laa The skew-symmetric orthogonal solutions of the matrix equation AX = B Chunjun Meng, Xiyan Hu, Lei Zhang College of Mathematics
More informationNecessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix.
arxiv:math/0506382v1 [math.na] 19 Jun 2005 Necessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix. Adviser: Charles R. Johnson Department of Mathematics College
More informationMATH 223 FINAL EXAM APRIL, 2005
MATH 223 FINAL EXAM APRIL, 2005 Instructions: (a) There are 10 problems in this exam. Each problem is worth five points, divided equally among parts. (b) Full credit is given to complete work only. Simply
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationFall 2016 MATH*1160 Final Exam
Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank
More informationMath 2174: Practice Midterm 1
Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationORIE 6300 Mathematical Programming I August 25, Recitation 1
ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Calvin Wylie Recitation 1 Scribe: Mateo Díaz 1 Linear Algebra Review 1 1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite)
More informationCalculation in the special cases n = 2 and n = 3:
9. The determinant The determinant is a function (with real numbers as values) which is defined for quadratic matrices. It allows to make conclusions about the rank and appears in diverse theorems and
More informationNew concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse
Lesson 10: Rank-nullity theorem, General solution of Ax = b (A 2 R mm ) New concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse Matrix rank. matrix nullity Denition. The
More informationMath 242 fall 2008 notes on problem session for week of This is a short overview of problems that we covered.
Math 242 fall 28 notes on problem session for week of 9-3-8 This is a short overview of problems that we covered.. For each of the following sets ask the following: Does it span R 3? Is it linearly independent?
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Recall: Parity of a Permutation S n the set of permutations of 1,2,, n. A permutation σ S n is even if it can be written as a composition of an
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationCPE 310: Numerical Analysis for Engineers
CPE 310: Numerical Analysis for Engineers Chapter 2: Solving Sets of Equations Ahmed Tamrawi Copyright notice: care has been taken to use only those web images deemed by the instructor to be in the public
More informationMA 511, Session 10. The Four Fundamental Subspaces of a Matrix
MA 5, Session The Four Fundamental Subspaces of a Matrix Let A be a m n matrix. (i) The row space C(A T )ofais the subspace of R n spanned by the rows of A. (ii) The null space N (A) ofa is the subspace
More informationRank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Rank and Nullity MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives We have defined and studied the important vector spaces associated with matrices (row space,
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationChapter 2: Matrix Algebra
Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry
More informationDM559 Linear and Integer Programming. Lecture 6 Rank and Range. Marco Chiarandini
DM559 Linear and Integer Programming Lecture 6 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 2 Outline 1. 2. 3. 3 Exercise Solve the
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe ECE133A (Winter 2018) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationMATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK 2 2.1 : 2, 5, 9, 12 2.3 : 3, 6 2.4 : 2, 4, 5, 9, 11 Section 2.1: Unitary Matrices Problem 2 If λ σ(u) and U M n is unitary, show that
More informationTopological Data Analysis - Spring 2018
Topological Data Analysis - Spring 2018 Simplicial Homology Slightly rearranged, but mostly copy-pasted from Harer s and Edelsbrunner s Computational Topology, Verovsek s class notes. Gunnar Carlsson s
More information3. The F Test for Comparing Reduced vs. Full Models. opyright c 2018 Dan Nettleton (Iowa State University) 3. Statistics / 43
3. The F Test for Comparing Reduced vs. Full Models opyright c 2018 Dan Nettleton (Iowa State University) 3. Statistics 510 1 / 43 Assume the Gauss-Markov Model with normal errors: y = Xβ + ɛ, ɛ N(0, σ
More informationRow Space and Column Space of a Matrix
Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationPart I True or False. (One point each. A wrong answer is subject to one point deduction.)
FACULTY OF ENGINEERING CHULALONGKORN UNIVERSITY 21121 Computer Engineering Mathematics YEAR II, Second Semester, Final Examination, March 3, 214, 13: 16: Name ID 2 1 CR58 Instructions 1. There are 43 questions,
More informationPractice Midterm 1 Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 2014
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 Student ID: Circle your section: Shin 8am 7 Evans Lim pm 35 Etcheverry Cho 8am 75 Evans 3 Tanzer pm 35 Evans 3 Shin
More informationELEMENTS OF MATRIX ALGEBRA
ELEMENTS OF MATRIX ALGEBRA CHUNG-MING KUAN Department of Finance National Taiwan University September 09, 2009 c Chung-Ming Kuan, 1996, 2001, 2009 E-mail: ckuan@ntuedutw; URL: homepagentuedutw/ ckuan CONTENTS
More information9. The determinant. Notation: Also: A matrix, det(a) = A IR determinant of A. Calculation in the special cases n = 2 and n = 3:
9. The determinant The determinant is a function (with real numbers as values) which is defined for square matrices. It allows to make conclusions about the rank and appears in diverse theorems and formulas.
More informationOn the Solution of Constrained and Weighted Linear Least Squares Problems
International Mathematical Forum, 1, 2006, no. 22, 1067-1076 On the Solution of Constrained and Weighted Linear Least Squares Problems Mohammedi R. Abdel-Aziz 1 Department of Mathematics and Computer Science
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationComputational Methods. Systems of Linear Equations
Computational Methods Systems of Linear Equations Manfred Huber 2010 1 Systems of Equations Often a system model contains multiple variables (parameters) and contains multiple equations Multiple equations
More informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationHUA S MATRIX EQUALITY AND SCHUR COMPLEMENTS
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 3, Number 1, Pages 1 18 c 2007 Institute for Scientific Computing and Information HUA S MATRIX EQUALITY AND SCHUR COMPLEMENTS CHRIS PAGE,
More information