Compound matrices and some classical inequalities

Size: px
Start display at page:

Download "Compound matrices and some classical inequalities"

Transcription

1 Compound matrices and some classical inequalities Tin-Yau Tam Mathematics & Statistics Auburn University Dec. 3, 04

2 We discuss some elegant proofs of several classical inequalities of matrices by using compound matrices: Weyl, Kostant, Yamamoto, Schur

3 I. Compound matrices A C n n, 1 k n. The kth compound of A is an ( ( ) n k) n k complex matrix Ck (A) whose elements are where α, β Q k,n, and C k (A) α,β = det A[α β], (1) Q k,n = {ω = (ω(1),..., ω(k)) : 1 ω(1) < < ω(k) n} Example: (1) k = 1, C 1 (A) = A, (2) k = n, C n (A) = det A. Example: n = 3 and k = 2 C 2 (A) = det A[1, 2 1, 2] det A[1, 2 1, 3] det A[1, 2 2, 3] det A[1, 3 1, 2] det A[1, 3 1, 3] det A[1, 3 2, 3] det A[2, 3 1, 2] det A[2, 3 1, 3] det A[2, 3 2, 3]

4 C k : GL n (C) GL ( n k )(C) is a group representation: C k (AB) = C k (A)C k (B) (Cauchy-Binet Formula) C k (I n ) = I ( n k ), (2) C k(a 1 ) = [C k (A)] 1 λ 1,..., λ n = the e-values of A in nonincreasing order λ 1 λ n The singular values of A are the nonnegative square roots of the e-values of the p.s.d. A A, also in nonincreasing order: α 1 α n. Remark: α 1 = max x 2 =1 Ax 2

5 Proposition 1: Let A C n n. 1. C k (A ) = [C k (A)]. Thus C k maps unitary matrices to unitary matrices. 2. The e-values of C k (A) are λ ω(j), ω Q k,n 3. The singular values of C k (A) are α ω(j), ω Q k,n 4. If A is upper triangular, so is C k (A) and the diagonal entries are a ω(j),ω(j), ω Q k,n

6 The kth additive compound of A: k (A) = d dt t=0 C k (I + ta), (2) or equivalently, C k (I + ta) = I + t k (A) + t 2 R + Example: If n = 3 and k = 2, then k (A) = a 11 + a 22 a 23 a 13 a 32 a 11 + a 33 a 12 a 31 a 21 a 22 + a 33. In general 1 (A) = A and n (A) = tr A. k : C n n C ( n k ) ( n k ) is an algebra representation of the Lie algebras equipped with the bracket operation [A, B] = AB BA. k = derivative of C k : GL n (C) GL ( n k )(C) at the identity.

7 Proposition 2: Let A C n n. 1. k (A ) = k (A). If A is Hermitian, so is k (A). 2. The e-values of k (A) are k λ ω(j), ω Q k,n. 3. The diagonal entries of k (A) are k a ω(j),ω(j), ω Q k,n

8 II. Weyl s Inequalities Weyl: Let A C n n. Then n λ j λ j = n α j, k = 1,..., n 1, (3) α j. (4) Proof: If x C n is a unit e-vector of A associated with λ 1, then Thus λ 1 = λ 1 x 2 = Ax 2 α 1. λ 1 α 1. (5) Recall α 1 α n and λ 1 λ n.

9 The largest singular value of C k (A) is α j The eigenvalue of maximum modulus of C k (A) is λ j Apply (5) on C k (A) and (3) follows. By considering the determinant of A, we have (4). Remark: Converse is true and is due to Horn. A generalization is true for real semisimple noncompact Lie groups (complete multiplicative Jordan decomposition).

10 III. Kostant s nonlinear convexity theorem A GL n (C), QR decomposition asserts: A = QR, Q U(n), R is upper with positive diagonal entries and the decomposition is unique. QR = Gram-Schmidt process Let a 1 a n > 0 (after rearrangement) be the diagonal entries of R. Each a i = distance between a column of A and the subspace spanned by the previous columns.

11 Kostant: Let A GL n (C). Then a j n a j = n α j, k = 1,..., n 1, (6) α j. (7) Proof: Suppose a 1 = the ith diagonal entry of R By the Gram-Schmidt process a 1 A i 2 where A i is the ith column of A. So a 1 α 1. (8) Notice that C k (A) = C k (Q)C k (R) is the QR-decomposition of C k (A)

12 The largest diagonal entry of C k (R) is a j. The largest singular value of C k (A) is k α j. Apply (8) to C k (A) to have (6). Determinant consideration leads to (7). Remark: The converse is true. The general result is about Iawasawa decomposition G = KAN of a noncompact semisimple Lie group G. Another proof: Q 1 A = R. Apply Weyl s Theorem to

13 IV. Yamamoto s theorem Classical result: lim m Am 1/m 2 = λ 1, (9) where A 2 = α 1 is the spectral norm of A. Yamamoto: Let A C n n. Then lim m [α i(a m )] 1/m = λ i, i = 1,..., n. (10) Proof: Recall α 1 α n, λ 1 λ n.

14 Apply (9) on the compound matrix C k (A): lim m [α 1(A m )] 1/m [α k (A m )] 1/m = lim m [α 1(A m ) α k (A m )] 1/m = lim m [α 1(C k (A m )] 1/m = λ 1 (C k (A)) = λ j, k = 1,..., n 1. Let 1 r n such that λ r 0 but λ r+1 = 0. From the above computation, lim m [α j(a m )] 1/m = λ j = 0, j = 1,..., r, and lim m [α j(a m )] 1/m = λ j = 0, j = r + 1,..., n. Remark: Yamamoto s theorem is very recently extended to noncompact Lie groups (KA + K decomposition).

15 V. Schur s inequalities Schur: Let A C n n be Hermitian with diagonal entries d 1 d n (after rearrangement) and e-values λ 1 λ n. Then k d j n d j = k n λ j, k = 1,..., n 1, (11) λ j. (12) Proof: By the spectral theorem for Hermitian matrices, there exists a unitary matrix U such that A = Udiag (λ 1,..., λ n )U. The diagonal of A is ( u ij 2 )λ, λ = (λ 1,..., λ n ) T.

16 The matrix ( u ij 2 ) is doubly stochastic, d 1 λ 1. (13) Applying (13) on k (A) and by Proposition 2, we get (11). By considering the trace of A, we have (12). Remark: The converse is true and is due to Horn. A generalization is true for any semisimple Lie algebra of a noncompact Lie group (Cartan decomposition and adjoint orbit). Another proof: Use interlacing inequalities for a Hermitian matrix and its principal submatrices.

AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX

AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX TIN-YAU TAM AND HUAJUN HUANG Abstract.

More information

AN ASYMPTOTIC BEHAVIOR OF QR DECOMPOSITION

AN ASYMPTOTIC BEHAVIOR OF QR DECOMPOSITION Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 AN ASYMPTOTIC BEHAVIOR OF QR DECOMPOSITION HUAJUN HUANG AND TIN-YAU TAM Abstract. The m-th root of the diagonal of the upper

More information

ALUTHGE ITERATION IN SEMISIMPLE LIE GROUP. 1. Introduction Given 0 < λ < 1, the λ-aluthge transform of X C n n [4]:

ALUTHGE ITERATION IN SEMISIMPLE LIE GROUP. 1. Introduction Given 0 < λ < 1, the λ-aluthge transform of X C n n [4]: Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 ALUTHGE ITERATION IN SEMISIMPLE LIE GROUP HUAJUN HUANG AND TIN-YAU TAM Abstract. We extend, in the context of connected noncompact

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

More information

Mathematics & Statistics Auburn University, Alabama, USA

Mathematics & Statistics Auburn University, Alabama, USA Mathematics & Statistics Auburn University, Alabama, USA May 25, 2011 On Bertram Kostant s paper: On convexity, the Weyl group and the Iwasawa decomposition Ann. Sci. École Norm. Sup. (4) 6 (1973), 413

More information

Convexity of the Joint Numerical Range

Convexity of the Joint Numerical Range Convexity of the Joint Numerical Range Chi-Kwong Li and Yiu-Tung Poon October 26, 2004 Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement. Abstract Let A = (A 1,..., A m ) be an

More information

Math 408 Advanced Linear Algebra

Math 408 Advanced Linear Algebra Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 4 Hermitian and symmetric matrices Basic properties Theorem Let A M n. The following are equivalent. Remark (a) A is Hermitian, i.e., A = A. (b) x

More information

ON THE QR ITERATIONS OF REAL MATRICES

ON THE QR ITERATIONS OF REAL MATRICES Unspecified Journal Volume, Number, Pages S????-????(XX- ON THE QR ITERATIONS OF REAL MATRICES HUAJUN HUANG AND TIN-YAU TAM Abstract. We answer a question of D. Serre on the QR iterations of a real matrix

More information

Conceptual Questions for Review

Conceptual 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 information

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS Abstract. We present elementary proofs of the Cauchy-Binet Theorem on determinants and of the fact that the eigenvalues of a matrix

More information

Spectral inequalities and equalities involving products of matrices

Spectral inequalities and equalities involving products of matrices Spectral inequalities and equalities involving products of matrices Chi-Kwong Li 1 Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187 (ckli@math.wm.edu) Yiu-Tung Poon Department

More information

EXAM. Exam 1. Math 5316, Fall December 2, 2012

EXAM. Exam 1. Math 5316, Fall December 2, 2012 EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.

More information

Chapter 3 Transformations

Chapter 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 information

Linear Algebra and its Applications

Linear Algebra and its Applications Linear Algebra and its Applications 432 (2010) 3250 3257 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Aluthge iteration in

More information

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

More information

Convex analysis on Cartan subspaces

Convex analysis on Cartan subspaces Nonlinear Analysis 42 (2000) 813 820 www.elsevier.nl/locate/na Convex analysis on Cartan subspaces A.S. Lewis ;1 Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario,

More information

NORMS ON SPACE OF MATRICES

NORMS ON SPACE OF MATRICES NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Ir 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 information

Diagonalizing Matrices

Diagonalizing Matrices Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B

More information

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian

2. 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 information

Algebra C Numerical Linear Algebra Sample Exam Problems

Algebra C Numerical Linear Algebra Sample Exam Problems Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric

More information

I. Multiple Choice Questions (Answer any eight)

I. Multiple Choice Questions (Answer any eight) Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY

More information

Mathematical Methods wk 2: Linear Operators

Mathematical Methods wk 2: Linear Operators John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm

More information

Lecture notes: Applied linear algebra Part 1. Version 2

Lecture notes: Applied linear algebra Part 1. Version 2 Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and

More information

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors Chapter 6 Eigenvalues and eigenvectors An eigenvalue of a square matrix represents the linear operator as a scaling of the associated eigenvector, and the action of certain matrices on general vectors

More information

LinGloss. A glossary of linear algebra

LinGloss. A glossary of linear algebra LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal

More information

Finite and infinite dimensional generalizations of Klyachko theorem. Shmuel Friedland. August 15, 1999

Finite and infinite dimensional generalizations of Klyachko theorem. Shmuel Friedland. August 15, 1999 Finite and infinite dimensional generalizations of Klyachko theorem Shmuel Friedland Department of Mathematics, Statistics, and Computer Science University of Illinois Chicago 322 SEO, 851 S. Morgan, Chicago,

More information

Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2. Review of Linear Algebra Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,

More information

Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

More information

Research Article A Note on Iwasawa-Type Decomposition

Research Article A Note on Iwasawa-Type Decomposition International Mathematics and Mathematical Sciences Volume 2011, Article ID 135167, 10 pages doi:101155/2011/135167 Research Article A Note on Iwasawa-Type Decomposition Philip Foth 1, 2 1 CEGEP, Champlain

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

Math 489AB Exercises for Chapter 2 Fall Section 2.3

Math 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 information

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

More information

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why? KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of

More information

The following definition is fundamental.

The following definition is fundamental. 1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic

More information

Section 6.4. The Gram Schmidt Process

Section 6.4. The Gram Schmidt Process Section 6.4 The Gram Schmidt Process Motivation The procedures in 6 start with an orthogonal basis {u, u,..., u m}. Find the B-coordinates of a vector x using dot products: x = m i= x u i u i u i u i Find

More information

Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence)

Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector

More information

Heinz-Kato s inequalities for semisimple Lie groups

Heinz-Kato s inequalities for semisimple Lie groups Journal of Lie Theory 0 (2009)???? c 2009 Heldermann Verlag Berlin Version of January 4, 2009 Heinz-Kato s inequalities for semisimple Lie groups Tin-Yau Tam Abstract. Extensions of Heinz-Kato s inequalities

More information

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe

More information

Throughout these notes we assume V, W are finite dimensional inner product spaces over C.

Throughout these notes we assume V, W are finite dimensional inner product spaces over C. Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal

More information

Lecture 7: Positive Semidefinite Matrices

Lecture 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 information

Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein

Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein Matrix Mathematics Theory, Facts, and Formulas with Application to Linear Systems Theory Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Special Symbols xv Conventions, Notation,

More information

Lecture notes on Quantum Computing. Chapter 1 Mathematical Background

Lecture notes on Quantum Computing. Chapter 1 Mathematical Background Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For

More information

1. General Vector Spaces

1. 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 information

Linear Algebra and Dirac Notation, Pt. 2

Linear Algebra and Dirac Notation, Pt. 2 Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14

More information

The Eigenvalue Problem: Perturbation Theory

The Eigenvalue Problem: Perturbation Theory Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just

More information

Linear algebra and applications to graphs Part 1

Linear algebra and applications to graphs Part 1 Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces

More information

2. Review of Linear Algebra

2. Review of Linear Algebra 2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA 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 information

Taxonomy of n n Matrices. Complex. Integer. Real. diagonalizable. Real. Doubly stochastic. Unimodular. Invertible. Permutation. Orthogonal.

Taxonomy of n n Matrices. Complex. Integer. Real. diagonalizable. Real. Doubly stochastic. Unimodular. Invertible. Permutation. Orthogonal. Doubly stochastic Taxonomy of n n Matrices Each rectangle represents one class of complex n n matrices. Arrows indicate subset relations. Classes in green are closed under multiplication. Classes in blue

More information

Some inequalities for sum and product of positive semide nite matrices

Some inequalities for sum and product of positive semide nite matrices Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,

More information

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue

More information

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.

More information

Topics in Applied Linear Algebra - Part II

Topics in Applied Linear Algebra - Part II Topics in Applied Linear Algebra - Part II April 23, 2013 Some Preliminary Remarks The purpose of these notes is to provide a guide through the material for the second part of the graduate module HM802

More information

Lecture 3: QR-Factorization

Lecture 3: QR-Factorization Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization

More information

Linear Algebra Lecture Notes-II

Linear Algebra Lecture Notes-II Linear Algebra Lecture Notes-II Vikas Bist Department of Mathematics Panjab University, Chandigarh-64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered

More information

QR decomposition: History and its Applications

QR decomposition: History and its Applications Mathematics & Statistics Auburn University, Alabama, USA Dec 17, 2010 decomposition: and its Applications Tin-Yau Tam èuî Æâ w f ŒÆêÆ ÆÆ Page 1 of 37 email: tamtiny@auburn.edu Website: www.auburn.edu/

More information

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

More information

Applied Linear Algebra

Applied Linear Algebra Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University

More information

Matrix Inequalities by Means of Block Matrices 1

Matrix Inequalities by Means of Block Matrices 1 Mathematical Inequalities & Applications, Vol. 4, No. 4, 200, pp. 48-490. Matrix Inequalities by Means of Block Matrices Fuzhen Zhang 2 Department of Math, Science and Technology Nova Southeastern University,

More information

Numerical Methods in Matrix Computations

Numerical Methods in Matrix Computations Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices

More information

Nonlinear Programming Algorithms Handout

Nonlinear Programming Algorithms Handout Nonlinear Programming Algorithms Handout Michael C. Ferris Computer Sciences Department University of Wisconsin Madison, Wisconsin 5376 September 9 1 Eigenvalues The eigenvalues of a matrix A C n n are

More information

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

Archive of past papers, solutions and homeworks for. MATH 224, Linear Algebra 2, Spring 2013, Laurence Barker

Archive of past papers, solutions and homeworks for. MATH 224, Linear Algebra 2, Spring 2013, Laurence Barker Archive of past papers, solutions and homeworks for MATH 224, Linear Algebra 2, Spring 213, Laurence Barker version: 4 June 213 Source file: archfall99.tex page 2: Homeworks. page 3: Quizzes. page 4: Midterm

More information

Fundamentals of Engineering Analysis (650163)

Fundamentals of Engineering Analysis (650163) Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

More information

Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras Canadian Mathematical Bulletin doi:10.4153/cmb-2010-097-7 c Canadian Mathematical Society 2010 Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras Wai-Shun Cheung and Tin-Yau Tam Abstract.

More information

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax = . (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

More information

MATH 581D FINAL EXAM Autumn December 12, 2016

MATH 581D FINAL EXAM Autumn December 12, 2016 MATH 58D FINAL EXAM Autumn 206 December 2, 206 NAME: SIGNATURE: Instructions: there are 6 problems on the final. Aim for solving 4 problems, but do as much as you can. Partial credit will be given on all

More information

Applied Linear Algebra in Geoscience Using MATLAB

Applied Linear Algebra in Geoscience Using MATLAB Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in

More information

Chapter 0 Miscellaneous Preliminaries

Chapter 0 Miscellaneous Preliminaries EE 520: Topics Compressed Sensing Linear Algebra Review Notes scribed by Kevin Palmowski, Spring 2013, for Namrata Vaswani s course Notes on matrix spark courtesy of Brian Lois More notes added by Namrata

More information

5 Compact linear operators

5 Compact linear operators 5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.

More information

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Name: TA Name and section: NO CALCULATORS, SHOW ALL WORK, NO OTHER PAPERS ON DESK. There is very little actual work to be done on this exam if

More information

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland

More information

LINEAR ALGEBRA SUMMARY SHEET.

LINEAR ALGEBRA SUMMARY SHEET. LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized

More information

October 25, 2013 INNER PRODUCT SPACES

October 25, 2013 INNER PRODUCT SPACES October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal

More information

Trace Inequalities for a Block Hadamard Product

Trace Inequalities for a Block Hadamard Product Filomat 32:1 2018), 285 292 https://doiorg/102298/fil1801285p Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Trace Inequalities for

More information

9.1 Eigenvectors and Eigenvalues of a Linear Map

9.1 Eigenvectors and Eigenvalues of a Linear Map Chapter 9 Eigenvectors and Eigenvalues 9.1 Eigenvectors and Eigenvalues of a Linear Map Given a finite-dimensional vector space E, letf : E! E be any linear map. If, by luck, there is a basis (e 1,...,e

More information

ME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ

ME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 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 information

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

Problem 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 information

Math 407: Linear Optimization

Math 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 information

Math 307 Learning Goals

Math 307 Learning Goals Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear

More information

Lecture 10 - Eigenvalues problem

Lecture 10 - Eigenvalues problem Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems

More information

Inner products and Norms. Inner product of 2 vectors. Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y x n y n in R n

Inner products and Norms. Inner product of 2 vectors. Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y x n y n in R n Inner products and Norms Inner product of 2 vectors Inner product of 2 vectors x and y in R n : x 1 y 1 + x 2 y 2 + + x n y n in R n Notation: (x, y) or y T x For complex vectors (x, y) = x 1 ȳ 1 + x 2

More information

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.

MATH 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 information

Applied Linear Algebra in Geoscience Using MATLAB

Applied Linear Algebra in Geoscience Using MATLAB Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in

More information

CANONICAL FORMS, HIGHER RANK NUMERICAL RANGES, TOTALLY ISOTROPIC SUBSPACES, AND MATRIX EQUATIONS

CANONICAL FORMS, HIGHER RANK NUMERICAL RANGES, TOTALLY ISOTROPIC SUBSPACES, AND MATRIX EQUATIONS CANONICAL FORMS, HIGHER RANK NUMERICAL RANGES, TOTALLY ISOTROPIC SUBSPACES, AND MATRIX EQUATIONS CHI-KWONG LI AND NUNG-SING SZE Abstract. Results on matrix canonical forms are used to give a complete description

More information

arxiv:math/ v1 [math.fa] 4 Jan 2007

arxiv:math/ v1 [math.fa] 4 Jan 2007 Tr[ABA] p = Tr[B 1/2 A 2 B 1/2 ] p. On the Araki-Lieb-Thirring inequality arxiv:math/0701129v1 [math.fa] 4 Jan 2007 Koenraad M.R. Audenaert Institute for Mathematical Sciences, Imperial College London

More information

Chapter 0 Preliminaries

Chapter 0 Preliminaries Chapter 0 Preliminaries Objective of the course Introduce basic matrix results and techniques that are useful in theory and applications. It is important to know many results, but there is no way I can

More information

Math 307 Learning Goals. March 23, 2010

Math 307 Learning Goals. March 23, 2010 Math 307 Learning Goals March 23, 2010 Course Description The course presents core concepts of linear algebra by focusing on applications in Science and Engineering. Examples of applications from recent

More information

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March

More information

Index. for generalized eigenvalue problem, butterfly form, 211

Index. for generalized eigenvalue problem, butterfly form, 211 Index ad hoc shifts, 165 aggressive early deflation, 205 207 algebraic multiplicity, 35 algebraic Riccati equation, 100 Arnoldi process, 372 block, 418 Hamiltonian skew symmetric, 420 implicitly restarted,

More information

The C-Numerical Range in Infinite Dimensions

The C-Numerical Range in Infinite Dimensions The C-Numerical Range in Infinite Dimensions Frederik vom Ende Technical University of Munich June 17, 2018 WONRA 2018 Joint work with Gunther Dirr (University of Würzburg) Frederik vom Ende (TUM) C-Numerical

More information

The Lusztig-Vogan Bijection in the Case of the Trivial Representation

The Lusztig-Vogan Bijection in the Case of the Trivial Representation The Lusztig-Vogan Bijection in the Case of the Trivial Representation Alan Peng under the direction of Guangyi Yue Department of Mathematics Massachusetts Institute of Technology Research Science Institute

More information

8. Diagonalization.

8. Diagonalization. 8. Diagonalization 8.1. Matrix Representations of Linear Transformations Matrix of A Linear Operator with Respect to A Basis We know that every linear transformation T: R n R m has an associated standard

More information

Problem # Max points possible Actual score Total 120

Problem # Max points possible Actual score Total 120 FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to

More information

7. Symmetric Matrices and Quadratic Forms

7. Symmetric Matrices and Quadratic Forms Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value

More information