Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach
|
|
- Jack Norton
- 5 years ago
- Views:
Transcription
1 Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach Bhaskar Ramasubramanian 1, Swanand R. Khare and Madhu N. Belur 3 December 17, Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA. Department of Mathematics, Indian Institute of Technology, Kharagpur, India. 3 Department of Electrical Engineering, Indian Institute of Technology Bombay, India. 1 / 13
2 Motivation / 13 Goal : Computing the minimal polynomial basis (MPB) of a given polynomial matrix. Current state of the art: Involves explicit knowledge of entries of polynomial matrix. Examples - matrix pencils, LQ factorization of Toeplitz matrices. Numerically robust algorithms the key. This work: Generic case - only work with degrees of entries in given polynomial matrix to determine degrees of entries in MPB. For specific case, get upper bound on degree structure of MPB. No numerical considerations since problem transformed to computing annihilator of a constant matrix.
3 Preliminaries - Minimal Polynomial Basis 3 / 13 R[s] - polynomials in s with real coefficients R m n [s] - m n matrix with entries from R[s]. (m < n) d - degree of polynomial vector, i.e., maximum among degrees of polynomial components. {q(s) R n [s] R(s)q(s) = 0} - nullspace of R(s) R m n [s]. Minimal Polynomial Basis Given R R m n [s], of degree d and rank m, let q 1, q,..., q n m be polynomially independent vectors in the nullspace of R arranged in non-decreasing order of their degrees a 1, a,..., a n m respectively. Then, the set {q 1, q,..., q n m } is an MPB if for any other set of n m polynomially independent vectors from the nullspace with degrees b 1 b b n m, it turns out that a i b i for i = 1,,..., n m. MPB is unique upto degrees of the polynomial vectors.
4 Preliminaries - Genericity of Parameters 4 / 13 Algebraic variety - set of solutions, E q R n to a system of polynomial equations. Zero equation in variables renders variety trivial. Nontrivial algebraic variety in R n is a set of measure zero. Genericity Property P in terms of variables p 1, p,..., p n R is said to be satisfied generically if the set of values p 1, p,..., p n that do not satisfy P form a non-trivial algebraic variety in R n. Examples Two nonzero polynomials are generically coprime. Entries of a square matrix chosen generically from R makes it nonsingular. Left primeness of a wide polynomial matrix.
5 Problem Formulation 5 / 13 Given R R m n [s], define D Z m n such that [D] ij := deg[r] ij. If [R] ij = 0, then [D] ij :=. Define the sets: Z + = {z Z z 0} Z + = Z + { } Can construct unique D Z m n + for given R R m n [s] This D is called the degree structure of R. Given D, there exist many R with the same degree structure. D(R) := {R R m n [s] with degree structure D Z m n + }
6 Problem Formulation, Main Result 6 / 13 Problem 1 Given R R m n [s] with degree structure D Z m n +, let Q R n (n m) [s] be such that the columns of Q form an MPB for the nullspace of R. Let K Z n (n m) + denote the degree structure of Q. Then, can we determine K from D? Problem Let R 1, R D(R) and Q 1, Q R n (n m) [s] be such that the columns of Q 1 and Q form MPBs for R 1 and R respectively. Let K 1 and K denote the degree structures of Q 1 and Q respectively. Then, is K 1 = K generically? Theorem 1 Given degree structures K 1, K of minimal polynomial bases corresponding to the same degree structure D, K 1 = K.
7 Toeplitz Matrices 7 / 13 Given R R m n [s] with degree d, R = R 0 + R 1 s + + R d s d where R i R m n for i = 0, 1,..., d. Construct a sequence of real structured matrices from the given polynomial matrix as: R 0 A R 1 A A 0 =.,A 1 =, A = 0 0 A 0 A 0, (1) 0 R d 0 0 A 0 where 0 s in the above equation are zero matrices of size m n and A i R (d+i+1)m (i+1)n. Stop when (d + i + 1)m (i + 1)n. (Right) kernels of A i related to nullspace of R.
8 Degree Structure of MPB of 1 3 Polynomial Matrix Let R R 1 3 [s] have degree structure D = [ a b c ] and let columns of Q R 3 [s] with deg struct K form an MPB of R. Assume: a b c; 0 / R; b c. Theorem For even c, the degree structure of the MPB is given by: c c K = c c () b c b c The degree structure of the MPB for odd c is given by K = c 1 c 1 c 1 (c b) c+1 c+1 c+1 (c b) (3) When c = b + k, the MPB will contain the zero polynomial, corresponding to a term in its degree structure. 8 / 13
9 Example 9 / 13 Given D = [ 0 1 ], find K = A 0 = ; A 1 =, such that DK = A 1 is a wide matrix. We have (D 0 + D 1 s + D s )K = 0, where D i corresponds to the coefficients of the degree i terms in the given polynomial matrix. Note: If a particular A i yields only some columns of the MPB, the remaining columns can be got by constructing A i+1.
10 Example 10 / 13 Need: Constant matrix P such that A 1 P = 0. For last row of A 1 to be annihilated, corresponding element(s) in K must be zero. This effectively eliminates the last column of A 1, as shown below: P = 0 0, yielding K =
11 Saturation 11 / 13 Observe: degree structure of K independent of a. Same algorithm can be used to determine degree structure of minimal left indices of K, D 1. If D D 1 (component wise), and they have an MPB with the same degree structure, K, then D is said to be saturated. Unsaturated D some degree of freedom to change one or more coefficients from 0 to a nonzero value,. Saturation: degree of freedom offered to replace zeros by nonzeros in degree structure of D while maintaining that of K. Proposition When D = [ a b c ] Z 1 3 +, and c b, D sat = [ b b c ]. D sat for higher dimensions of D? Not (yet) known.
12 Conclusions 1 / 13 Degree structure of MPB of a given polynomial matrix depends only on its degree structure, and not its coefficients. Exact degree structure of MPB got for generic case; same serves as upper bound for specific case. MPB of 1 3 degree structure studied in detail. Closed form solution for degree structure of MPB provided. Genericity of parameters ensured matrices had full rank. Saturation of a degree structure examined in the context of freedom of making some zero coefficients nonzero. Complete absence of extensive numerical computations - results depended only on zero/ nonzero nature of coefficients.
13 Thank You 13 / 13
Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach
Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach Bhaskar Ramasubramanian, Swanand R Khare and Madhu N Belur Abstract This paper formulates the problem
More informationStructural Resilience of Cyberphysical Systems Under Attack
Structural Resilience of Cyberphysical Systems Under Attack Bhaskar Ramasubramanian 1, M.A. Rajan 2, M. Girish Chandra 2 1 Department of Electrical and Computer Engineering, and Institute for Systems Research,
More informationAlgorithm to Compute Minimal Nullspace Basis of a Polynomial Matrix
Proceedings of the 19th International Symposium on Mathematical heory of Networks and Systems MNS 1 5 9 July, 1 Budapest, Hungary Algorithm to Compute Minimal Nullspace Basis of a Polynomial Matrix S.
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
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 informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationLinear Equations in Linear Algebra
Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p
More informationChapter 1. Vectors, Matrices, and Linear Spaces
1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution
More informationProduct distance matrix of a tree with matrix weights
Product distance matrix of a tree with matrix weights R B Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India email: rbb@isidacin Sivaramakrishnan Sivasubramanian
More informationx y B =. v u Note that the determinant of B is xu + yv = 1. Thus B is invertible, with inverse u y v x On the other hand, d BA = va + ub 2
5. Finitely Generated Modules over a PID We want to give a complete classification of finitely generated modules over a PID. ecall that a finitely generated module is a quotient of n, a free module. Let
More informationEE263: Introduction to Linear Dynamical Systems Review Session 2
EE263: Introduction to Linear Dynamical Systems Review Session 2 Basic concepts from linear algebra nullspace range rank and conservation of dimension EE263 RS2 1 Prerequisites We assume that you are familiar
More informationNumerical Linear Algebra Homework Assignment - Week 2
Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationLinear Algebra I Lecture 8
Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019 Outline 1 2 Gauss-Jordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
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 information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationTest 3, Linear Algebra
Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationLecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases.
Lecture 11 Andrei Antonenko February 6, 003 1 Examples of bases Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Example 1.1. Consider the vector space P the
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationSingularity of the discrete Laplacian operator
Singularity of the discrete Laplacian operator Andrea Alessandro Ruggiu Jan Nordström Singularity of the Laplacian A.A.Ruggiu, J.Nordström August 24, 2015 1 Motivation In the SBP SAT framework, for unsteady
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 02 Vector Spaces, Subspaces, linearly Dependent/Independent of
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 informationLinear Algebra V = T = ( 4 3 ).
Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional
More informationSplitting Subspaces, Singer Cycles and Linear Recurrences
Splitting Subspaces, Singer Cycles and Linear Recurrences Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076, India http://www.math.iitb.ac.in/ srg/ Séminaire de Théorie
More informationChapter 6 - Orthogonality
Chapter 6 - Orthogonality Maggie Myers Robert A. van de Geijn The University of Texas at Austin Orthogonality Fall 2009 http://z.cs.utexas.edu/wiki/pla.wiki/ 1 Orthogonal Vectors and Subspaces http://z.cs.utexas.edu/wiki/pla.wiki/
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationLECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK)
LECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK) In this lecture, F is a fixed field. One can assume F = R or C. 1. More about the spanning set 1.1. Let S = { v 1, v n } be n vectors in V, we have defined
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. #07 Jordan Canonical Form Cayley Hamilton Theorem (Refer Slide Time:
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n
More informationLinear maps. Matthew Macauley. Department of Mathematical Sciences Clemson University Math 8530, Spring 2017
Linear maps Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Spring 2017 M. Macauley (Clemson) Linear maps Math 8530, Spring 2017
More information(i) [7 points] Compute the determinant of the following matrix using cofactor expansion.
Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det
More informationLinear Independence x
Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationLinear Algebra. Session 8
Linear Algebra. Session 8 Dr. Marco A Roque Sol 08/01/2017 Abstract Linear Algebra Range and kernel Let V, W be vector spaces and L : V W, be a linear mapping. Definition. The range (or image of L is the
More informationAlgebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix
Algebra & Trig. I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural
More informationRow Reduction and Echelon Forms
Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence
More informationOn Systems of Diagonal Forms II
On Systems of Diagonal Forms II Michael P Knapp 1 Introduction In a recent paper [8], we considered the system F of homogeneous additive forms F 1 (x) = a 11 x k 1 1 + + a 1s x k 1 s F R (x) = a R1 x k
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More information3. Replace any row by the sum of that row and a constant multiple of any other row.
Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
More informationDepartment of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016
Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More information9. Numerical linear algebra background
Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationMath Final December 2006 C. Robinson
Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationLetting be a field, e.g., of the real numbers, the complex numbers, the rational numbers, the rational functions W(s) of a complex variable s, etc.
1 Polynomial Matrices 1.1 Polynomials Letting be a field, e.g., of the real numbers, the complex numbers, the rational numbers, the rational functions W(s) of a complex variable s, etc., n ws ( ) as a
More informationAlgebraic Methods. Motivation: Systems like this: v 1 v 2 v 3 v 4 = 1 v 1 v 2 v 3 v 4 = 0 v 2 v 4 = 0
Motivation: Systems like this: v v 2 v 3 v 4 = v v 2 v 3 v 4 = 0 v 2 v 4 = 0 are very difficult for CNF SAT solvers although they can be solved using simple algebraic manipulations Let c 0, c,...,c 2 n
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 informationThe inverse of a tridiagonal matrix
Linear Algebra and its Applications 325 (2001) 109 139 www.elsevier.com/locate/laa The inverse of a tridiagonal matrix Ranjan K. Mallik Department of Electrical Engineering, Indian Institute of Technology,
More informationMatrices and RRE Form
Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that
More informationChapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.
Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]
More informationDetermine whether the following system has a trivial solution or non-trivial solution:
Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding
More informationAFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION
4. BASES AND DIMENSION Definition Let u 1,..., u n be n vectors in V. The vectors u 1,..., u n are linearly independent if the only linear combination of them equal to the zero vector has only zero scalars;
More informationA 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2
46 MATH 223A NOTES 2011 LIE ALGEBRAS 11. Classification of semisimple Lie algebras I will explain how the Cartan matrix and Dynkin diagrams describe root systems. Then I will go through the classification
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationALGEBRAIC GEOMETRY I - FINAL PROJECT
ALGEBRAIC GEOMETRY I - FINAL PROJECT ADAM KAYE Abstract This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over C Following the technique of Ryan [3] for
More informationMath 240 Calculus III
Generalized Calculus III Summer 2015, Session II Thursday, July 23, 2015 Agenda 1. 2. 3. 4. Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to make
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationMath 54. Selected Solutions for Week 5
Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3
More information( 9x + 3y. y 3y = (λ 9)x 3x + y = λy 9x + 3y = 3λy 9x + (λ 9)x = λ(λ 9)x. (λ 2 10λ)x = 0
Math 46 (Lesieutre Practice final ( minutes December 9, 8 Problem Consider the matrix M ( 9 a Prove that there is a basis for R consisting of orthonormal eigenvectors for M This is just the spectral theorem:
More informationLecture 11: Finish Gaussian elimination and applications; intro to eigenvalues and eigenvectors (1)
Lecture 11: Finish Gaussian elimination and applications; intro to eigenvalues and eigenvectors (1) Travis Schedler Tue, Oct 18, 2011 (version: Tue, Oct 18, 6:00 PM) Goals (2) Solving systems of equations
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 informationStabilization, Pole Placement, and Regular Implementability
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002 735 Stabilization, Pole Placement, and Regular Implementability Madhu N. Belur and H. L. Trentelman, Senior Member, IEEE Abstract In this
More informationMath "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25
Math 102 6.3 "Matrix Approach to Solving Systems" Bibiana Lopez Crafton Hills College November 2010 (CHC) 6.3 November 2010 1 / 25 Objectives: * Define a matrix and determine its order. * Write the augmented
More informationMAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :
MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..
More informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationSystem of Linear Equations
Chapter 7 - S&B Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination-
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 informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More information1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic
1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What
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 information2.57 PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA) The Fundamental Theorem of Algebra (FTA)
2.57 PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA) The Fundamental Theorem of Algebra (FTA) If f ( x) is a nonconstant n th -degree polynomial in standard form with real coefficients, then it must have
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
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 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 informationInfinite elementary divisor structure-preserving transformations for polynomial matrices
Infinite elementary divisor structure-preserving transformations for polynomial matrices N P Karampetakis and S Vologiannidis Aristotle University of Thessaloniki, Department of Mathematics, Thessaloniki
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationa. Define your variables. b. Construct and fill in a table. c. State the Linear Programming Problem. Do Not Solve.
Math Section. Example : The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 4 students, requires chaperones, and costs $, to rent. Each
More informationLecture 12: Solving Systems of Linear Equations by Gaussian Elimination
Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.
More informationComputing Minimal Nullspace Bases
Computing Minimal Nullspace ases Wei Zhou, George Labahn, and Arne Storjohann Cheriton School of Computer Science University of Waterloo, Waterloo, Ontario, Canada {w2zhou,glabahn,astorjoh}@uwaterloo.ca
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationStructured Matrices and Solving Multivariate Polynomial Equations
Structured Matrices and Solving Multivariate Polynomial Equations Philippe Dreesen Kim Batselier Bart De Moor KU Leuven ESAT/SCD, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium. Structured Matrix Days,
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationLinear equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
More informationIntroduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
More informationOrthogonality. 6.1 Orthogonal Vectors and Subspaces. Chapter 6
Chapter 6 Orthogonality 6.1 Orthogonal Vectors and Subspaces Recall that if nonzero vectors x, y R n are linearly independent then the subspace of all vectors αx + βy, α, β R (the space spanned by x and
More informationUse of Scilab to demonstrate concepts in linear algebra and polynomials
Use of Scilab to demonstrate concepts in linear algebra and polynomials Dr. Madhu N. Belur Control & Computing Department of Electrical Engineering Indian Institute of Technology Bombay Email: belur@ee.iitb.ac.in
More informationLecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued
Definition The process of solving a system of linear equations by converting the system to an augmented matrix is called Gaussian Elimination. The general strategy is as follows: Convert the system of
More information(II.B) Basis and dimension
(II.B) Basis and dimension How would you explain that a plane has two dimensions? Well, you can go in two independent directions, and no more. To make this idea precise, we formulate the DEFINITION 1.
More informationHASSE-MINKOWSKI THEOREM
HASSE-MINKOWSKI THEOREM KIM, SUNGJIN 1. Introduction In rough terms, a local-global principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally.
More information