Lecture 22: Jordan canonical form of upper-triangular matrices (1)
|
|
- Diane Atkinson
- 6 years ago
- Views:
Transcription
1 Lecture 22: Jordan canonical form of upper-triangular matrices (1) Travis Schedler Tue, Dec 6, 2011 (version: Tue, Dec 6, 1:00 PM)
2 Goals (2) Definition, existence, and uniqueness of Jordan canonical form of upper-triangular matrices How to compute Jordan form Relation with the minimal and characteristic polynomials Read Chapters 8 and 10, do PS 11.
3 Warm-up: uniqueness of Jordan form (3) (a) Let N n be the nilpotent, n n upper-triangular matrix N n = Compute dim null(nn k ) for all k 1. (b) Suppose that A is block diagonal, with diagonal blocks N ni for some n i 1. Using (a), show that dim(null(a k )) dim(null(a k 1 )) = # blocks N ni with n i k, where null(a 0 ) := null(i ) = {0}. (c) Conclude from (b) that, k, # blocks N k is (dim(null(a k )) dim(null(a k 1 ))) (dim(null(a k+1 )) dim(null(a k ))).
4 Solution to warm-up (4) (a) The matrix N k n is the same as N n but with the 1 s appearing a distance k from the main diagonal (horizontally or vertically), rather than distance 1 (unless k n). So the first k columns of N k n are zero, and the next n k are linearly independent. Hence dim null(n k n ) = min(k, n). (b) dim(null(nn k )) dim(null(nn k 1 )) = 1 if n k, and 0 if n < k. Now, dim(null(a k )) = sum of dim(null(nn k i )) over all diagonal blocks N ni. So dim(null(a k )) dim(null(a k 1 )) = # of blocks N ni with n i k. (c) The number of blocks with n i k minus the number of blocks with n i k + 1 is the number of blocks with n i = k exactly.
5 Jordan canonical form (5) Theorem If T admits a basis in which M(T ) is upper-triangular, then it admits a basis in which M(T ) is block-diagonal with blocks λ 1 0 λ 1 λi + N n = λ Such a matrix is called Jordan canonical form. It is unique for T up to rearranging the order of the blocks. Corollary: Over C, every T can be put in Jordan form. Corollary: Over C, two matrices are conjugate iff they have the same Jordan canonical form (up to permuting blocks). More generally, this applies over any F, to matrices which are conjugate to upper-triangular ones.
6 Examples of and how to compute Jordan form (6) If T admits an eigenbasis [orthonormal or not!] then its Jordan form is diagonal, and conversely. If T : R 2 R 2 is the rotation, its ( Jordan form ) doesn t exist; 0 1 the corresponding complex matrix has Jordan form 1 0 ( i 0 0 i ). Upper-triangular examples, on the board! E.g., How to compute in general: For each eigenvalue λ, find V (λ), compute N = (T λi ) V (λ), find its Jordan form (e.g., compute dim null(n k ) for all k 0).
7 Fun applications of Jordan form (7) Important: compute functions of matrices: exp(a), sin(a), log(a), etc. For a diagonal matrix: Just take the function on each diagonal entry. For diagonalizable, can do it in a basis where the matrix is diagonal. Also, given a Taylor series at 0, e.g., 1 + x + x 2 /2! +, can plug in a nilpotent matrix and get a finite sum. For a Jordan nilpotent matrix, N k is easy to write! Get immediately the answer. For λi + N, can still take exp(λi + N) = exp(λi ) exp(n) = e λ exp(n). For general f (x), on each V (λ), take Taylor series of f (x) at x = λ and plug in T V (λ) : finite sum again, get f (T ) V (λ). We did this on PS 10 for f (x) = x (Taylor series at x = 1)! Works for F = C, and for T which admit upper-tri matrices!
8 Solving differential equations (8) For example: a system v (x) = Av(x), for A Mat(C, n, n), v(x) Mat(C, n, 1) for all x (function of x: Solution: v(x) = e Ax. Now we know how to compute this! On each V (λ), e Ax becomes e (λi +N)x = e λx e Nx. This also computes solutions to constant-coefficient ODEs f (n) = a 0 f + a 1 f + + a n 1 f (n 1) : Write v = (f, f,..., f (n 1) ) t and then the system is the same as v (x) = Av(x), a a 1 A = a n a n 1
9 Uniqueness of Jordan canonical form (9) Any Jordan matrix is obtained from bases of each of the V (λ). So, it suffices to assume there is only one eigenvalue λ. Up to replacing T with T λi, we can assume T = N is nilpotent. Now, we need only show that the sizes n i that appear, and the number of times each appears, are independent of the basis. This follows from the warm-up exercise! (The number of times n i = k appears is dim null(a k+1 ) 2 dim null(a k ) + dim null(a k 1 )).
10 Proof of Jordan canonical form (10) Jordan canonical form is based on the following key result. Let N L(V ) be nilpotent and for nonzero v V, let m(v) be the maximum nonnegative integer such that N m(v) v 0. Lemma (Lemma 8.40) There exist vectors v 1,..., v k V such that (v 1, Nv 1,..., N m(v 1) v 1,..., v k, Nv k,..., N m(v k) v k ) is a basis of V. Note: the condition implies that (N m(v 1) v 1,..., N m(v k) v k ) is a basis of null N (consider any linear combination sent to zero by N). We prove by induction on dim V. Since range N V, we can assume that N range N has such vectors u 1,..., u j. Since u 1,..., u j range N, we can pick v 1,..., v j with Nv i = u i. Furthermore, let v j+1,..., v k be vectors which extend (N m(u 1) u 1,..., N m(u j ) u j ) to a basis of null N.
11 Completion of proof of the lemma (11) Claim: v 1,..., v k give a basis of the desired form. We show linear independence. Suppose i,j a i,jn j v i = 0. Applying N yields i,j a i,jn j u i = 0. By assumption that the u i give a basis of range N, we have a i,j = 0 whenever j m(u i ) = m(v i ) 1. Thus the only nonzero coefficients are those of (N m(v 1) v 1,..., N m(v k) v k ). These form a basis of null N. So all the coefficients are zero. Now, the length of the linearly independent list is dim range N + k = dim range N + dim null N = dim V, so it must be a basis.
12 Existence of Jordan canonical form (Theorem 8.47) (12) In the reverse of the basis of the lemma, M(N) is block diagonal with k blocks of sizes m(v k ),..., m(v 1 ), each of the form Now, for a general operator T, we can choose a basis as above for each V (λ), so that the nilpotent operator (T λi ) V (λ) has the above form. Putting them together, M(T ) is in Jordan form.
13 Char. and min. polys of Jordan matrices (13) A Jordan matrix is upper-triangular, so the char. poly. is λ (x λ)d λ, d λ = # of times λ is on the diagonal. For each Jordan block J = λi + N k, p(j) = 0 if and only if (x λ) k p(x). So, the minimal polynomial is λ (x λ)m λ, where m λ = the maximum size of Jordan block with eigenvalue (diag. entry) λ.
Lecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1)
Lecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1) Travis Schedler Tue, Nov 29, 2011 (version: Tue, Nov 29, 1:00 PM) Goals
More informationTravis Schedler. Thurs, Oct 27, 2011 (version: Thurs, Oct 27, 1:00 PM)
Lecture 13: Proof of existence of upper-triangular matrices for complex linear transformations; invariant subspaces and block upper-triangular matrices for real linear transformations (1) Travis Schedler
More informationLecture 23: Determinants (1)
Lecture 23: Determinants (1) Travis Schedler Thurs, Dec 8, 2011 (version: Thurs, Dec 8, 9:35 PM) Goals (2) Warm-up: minimal and characteristic polynomials of Jordan form matrices Snapshot: Generalizations
More informationLecture 19: Polar and singular value decompositions; generalized eigenspaces; the decomposition theorem (1)
Lecture 19: Polar and singular value decompositions; generalized eigenspaces; the decomposition theorem (1) Travis Schedler Thurs, Nov 17, 2011 (version: Thurs, Nov 17, 1:00 PM) Goals (2) Polar decomposition
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 19: Polar and singular value decompositions; generalized eigenspaces; the decomposition theorem (1)
Lecture 19: Polar and singular value decompositions; generalized eigenspaces; the decomposition theorem (1) Travis Schedler Thurs, Nov 17, 2011 (version: Thurs, Nov 17, 1:00 PM) Goals (2) Polar decomposition
More informationLecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1)
Lecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1) Travis Schedler Thurs, Nov 18, 2010 (version: Wed, Nov 17, 2:15
More informationMath 121 Practice Final Solutions
Math Practice Final Solutions December 9, 04 Email me at odorney@college.harvard.edu with any typos.. True or False. (a) If B is a 6 6 matrix with characteristic polynomial λ (λ ) (λ + ), then rank(b)
More informationMath 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler.
Math 453 Exam 3 Review The syllabus for Exam 3 is Chapter 6 (pages -2), Chapter 7 through page 37, and Chapter 8 through page 82 in Axler.. You should be sure to know precise definition of the terms we
More informationEIGENVALUES AND EIGENVECTORS 3
EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices
More informationFALL 2011, SOLUTION SET 10 LAST REVISION: NOV 27, 9:45 AM. (T c f)(x) = f(x c).
18.700 FALL 2011, SOLUTION SET 10 LAST REVISION: NOV 27, 9:45 AM TRAVIS SCHEDLER (1) Let V be the vector space of all continuous functions R C. For all c R, let T c L(V ) be the shift operator, which sends
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMath 108b: Notes on the Spectral Theorem
Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator
More informationLecture 23: Trace and determinants! (1) (Final lecture)
Lecture 23: Trace and determinants! (1) (Final lecture) Travis Schedler Thurs, Dec 9, 2010 (version: Monday, Dec 13, 3:52 PM) Goals (2) Recall χ T (x) = (x λ 1 ) (x λ n ) = x n tr(t )x n 1 + +( 1) n det(t
More information1.4 Solvable Lie algebras
1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors
ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 34 The powers of matrix Consider the following dynamic
More informationNOTES II FOR 130A JACOB STERBENZ
NOTES II FOR 130A JACOB STERBENZ Abstract. Here are some notes on the Jordan canonical form as it was covered in class. Contents 1. Polynomials 1 2. The Minimal Polynomial and the Primary Decomposition
More informationDiagonalization of Matrix
of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that
More informationLIE ALGEBRAS: LECTURE 3 6 April 2010
LIE ALGEBRAS: LECTURE 3 6 April 2010 CRYSTAL HOYT 1. Simple 3-dimensional Lie algebras Suppose L is a simple 3-dimensional Lie algebra over k, where k is algebraically closed. Then [L, L] = L, since otherwise
More informationJordan blocks. Defn. Let λ F, n Z +. The size n Jordan block with e-value λ is the n n upper triangular matrix. J n (λ) =
Jordan blocks Aim lecture: Even over F = C, endomorphisms cannot always be represented by a diagonal matrix. We give Jordan s answer, to what is the best form of the representing matrix. Defn Let λ F,
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 informationAgenda: Understand the action of A by seeing how it acts on eigenvectors.
Eigenvalues and Eigenvectors If Av=λv with v nonzero, then λ is called an eigenvalue of A and v is called an eigenvector of A corresponding to eigenvalue λ. Agenda: Understand the action of A by seeing
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationLecture 12: Diagonalization
Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors
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 informationREU 2007 Apprentice Class Lecture 8
REU 2007 Apprentice Class Lecture 8 Instructor: László Babai Scribe: Ian Shipman July 5, 2007 Revised by instructor Last updated July 5, 5:15 pm A81 The Cayley-Hamilton Theorem Recall that for a square
More informationICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization
ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 21 Symmetric matrices An n n
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationLecture 6: Corrections; Dimension; Linear maps
Lecture 6: Corrections; Dimension; Linear maps Travis Schedler Tues, Sep 28, 2010 (version: Tues, Sep 28, 4:45 PM) Goal To briefly correct the proof of the main Theorem from last time. (See website for
More informationMATH SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER Given vector spaces V and W, V W is the vector space given by
MATH 110 - SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER 2009 GSI: SANTIAGO CAÑEZ 1. Given vector spaces V and W, V W is the vector space given by V W = {(v, w) v V and w W }, with addition and scalar
More informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
More informationGeneralized eigenspaces
Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More informationLinear Algebra 1. M.T.Nair Department of Mathematics, IIT Madras. and in that case x is called an eigenvector of T corresponding to the eigenvalue λ.
Linear Algebra 1 M.T.Nair Department of Mathematics, IIT Madras 1 Eigenvalues and Eigenvectors 1.1 Definition and Examples Definition 1.1. Let V be a vector space (over a field F) and T : V V be a linear
More informationMath 489AB Exercises for Chapter 2 Fall Section 2.3
Math 489AB Exercises for Chapter 2 Fall 2008 Section 2.3 2.3.3. Let A M n (R). Then the eigenvalues of A are the roots of the characteristic polynomial p A (t). Since A is real, p A (t) is a polynomial
More informationMath 113 Final Exam: Solutions
Math 113 Final Exam: Solutions Thursday, June 11, 2013, 3.30-6.30pm. 1. (25 points total) Let P 2 (R) denote the real vector space of polynomials of degree 2. Consider the following inner product on P
More informationThe Jordan Canonical Form
The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finite-dimensional vector space over an algebraically closed field. Here we develop
More informationHomework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then
Homework 6 Solutions 1 Let V be the real vector space spanned by the functions e t, te t, t 2 e t, e 2t Find a Jordan canonical basis and a Jordan canonical form of T on V dened by T (f) = f Solution Note
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationLecture 15, 16: Diagonalization
Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose
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 informationSolution. That ϕ W is a linear map W W follows from the definition of subspace. The map ϕ is ϕ(v + W ) = ϕ(v) + W, which is well-defined since
MAS 5312 Section 2779 Introduction to Algebra 2 Solutions to Selected Problems, Chapters 11 13 11.2.9 Given a linear ϕ : V V such that ϕ(w ) W, show ϕ induces linear ϕ W : W W and ϕ : V/W V/W : Solution.
More informationMath 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures
More informationJordan Normal Form. Chapter Minimal Polynomials
Chapter 8 Jordan Normal Form 81 Minimal Polynomials Recall p A (x) =det(xi A) is called the characteristic polynomial of the matrix A Theorem 811 Let A M n Then there exists a unique monic polynomial q
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 informationFundamental theorem of modules over a PID and applications
Fundamental theorem of modules over a PID and applications Travis Schedler, WOMP 2007 September 11, 2007 01 The fundamental theorem of modules over PIDs A PID (Principal Ideal Domain) is an integral domain
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationA NOTE ON THE JORDAN CANONICAL FORM
A NOTE ON THE JORDAN CANONICAL FORM H. Azad Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia hassanaz@kfupm.edu.sa Abstract A proof of the Jordan
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationThe Cyclic Decomposition of a Nilpotent Operator
The Cyclic Decomposition of a Nilpotent Operator 1 Introduction. J.H. Shapiro Suppose T is a linear transformation on a vector space V. Recall Exercise #3 of Chapter 8 of our text, which we restate here
More informationThe eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute
A. [ 3. Let A = 5 5 ]. Find all (complex) eigenvalues and eigenvectors of The eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute 3 λ A λi =, 5 5 λ from which det(a λi)
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
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 informationMath Matrix Algebra
Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
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 information3 (Maths) Linear Algebra
3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra
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 information5.3.5 The eigenvalues are 3, 2, 3 (i.e., the diagonal entries of D) with corresponding eigenvalues. Null(A 3I) = Null( ), 0 0
535 The eigenvalues are 3,, 3 (ie, the diagonal entries of D) with corresponding eigenvalues,, 538 The matrix is upper triangular so the eigenvalues are simply the diagonal entries, namely 3, 3 The corresponding
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationLINEAR ALGEBRA MICHAEL PENKAVA
LINEAR ALGEBRA MICHAEL PENKAVA 1. Linear Maps Definition 1.1. If V and W are vector spaces over the same field K, then a map λ : V W is called a linear map if it satisfies the two conditions below: (1)
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin
More informationMath 25a Practice Final #1 Solutions
Math 25a Practice Final #1 Solutions Problem 1. Suppose U and W are subspaces of V such that V = U W. Suppose also that u 1,..., u m is a basis of U and w 1,..., w n is a basis of W. Prove that is a basis
More informationJordan normal form notes (version date: 11/21/07)
Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let
More informationA Brief Outline of Math 355
A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting
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 informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationSpectral radius, symmetric and positive matrices
Spectral radius, symmetric and positive matrices Zdeněk Dvořák April 28, 2016 1 Spectral radius Definition 1. The spectral radius of a square matrix A is ρ(a) = max{ λ : λ is an eigenvalue of A}. For an
More informationLecture 11: Diagonalization
Lecture 11: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 11 1 / 11 Definition The n n matrix A is diagonalizableif there exits nonsingular matrix P d 1 0 0. such that P 1 AP = D, where
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 informationDefinition (T -invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationUniversity of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm
University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm Name: The proctor will let you read the following conditions
More informationMath 489AB Exercises for Chapter 1 Fall Section 1.0
Math 489AB Exercises for Chapter 1 Fall 2008 Section 1.0 1.0.2 We want to maximize x T Ax subject to the condition x T x = 1. We use the method of Lagrange multipliers. Let f(x) = x T Ax and g(x) = x T
More informationJordan Canonical Form of a Nilpotent Matrix. Math 422
Jordan Canonical Form of a Nilpotent Matrix Math Schur s Triangularization Theorem tells us that every matrix A is unitarily similar to an upper triangular matrix T However, the only thing certain at this
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationc 1 v 1 + c 2 v 2 = 0 c 1 λ 1 v 1 + c 2 λ 1 v 2 = 0
LECTURE LECTURE 2 0. Distinct eigenvalues I haven t gotten around to stating the following important theorem: Theorem: A matrix with n distinct eigenvalues is diagonalizable. Proof (Sketch) Suppose n =
More informationLINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS
LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in
More informationEigenvalues and Eigenvectors
November 3, 2016 1 Definition () The (complex) number λ is called an eigenvalue of the n n matrix A provided there exists a nonzero (complex) vector v such that Av = λv, in which case the vector v is called
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationPositive entries of stable matrices
Positive entries of stable matrices Shmuel Friedland Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago Chicago, Illinois 60607-7045, USA Daniel Hershkowitz,
More informationOctober 4, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
October 4, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationThe Cayley-Hamilton Theorem and the Jordan Decomposition
LECTURE 19 The Cayley-Hamilton Theorem and the Jordan Decomposition Let me begin by summarizing the main results of the last lecture Suppose T is a endomorphism of a vector space V Then T has a minimal
More informationMATRICES ARE SIMILAR TO TRIANGULAR MATRICES
MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 4 Apr. 4, 2018 1 / 40 Recap Vector space, linear space, linear vector space Subspace Linearly independence and dependence Dimension, Basis, Change of Basis 2 / 40
More informationChapter 4 Euclid Space
Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,
More information(Can) Canonical Forms Math 683L (Summer 2003) M n (F) C((x λ) ) =
(Can) Canonical Forms Math 683L (Summer 2003) Following the brief interlude to study diagonalisable transformations and matrices, we must now get back to the serious business of the general case. In this
More informationMATH JORDAN FORM
MATH 53 JORDAN FORM Let A,, A k be square matrices of size n,, n k, respectively with entries in a field F We define the matrix A A k of size n = n + + n k as the block matrix A 0 0 0 0 A 0 0 0 0 A k It
More informationNotes on the matrix exponential
Notes on the matrix exponential Erik Wahlén erik.wahlen@math.lu.se February 14, 212 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential e A when A is not
More informationMath 113 Homework 5. Bowei Liu, Chao Li. Fall 2013
Math 113 Homework 5 Bowei Liu, Chao Li Fall 2013 This homework is due Thursday November 7th at the start of class. Remember to write clearly, and justify your solutions. Please make sure to put your name
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 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 informationSeptember 26, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
September 26, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationMath 113 Winter 2013 Prof. Church Midterm Solutions
Math 113 Winter 2013 Prof. Church Midterm Solutions Name: Student ID: Signature: Question 1 (20 points). Let V be a finite-dimensional vector space, and let T L(V, W ). Assume that v 1,..., v n is a basis
More information