MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006

Size: px
Start display at page:

Download "MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006"

Transcription

1 MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK : 2, 5, 9, : 3, : 2, 4, 5, 9, 11 Section 2.1: Unitary Matrices Problem 2 If λ σ(u) and U M n is unitary, show that λ = 1. Solution. If λ σ(u), U M n is unitary, and Ux = λx for x 0, then by Theorem 2.1.4(g), we have x C n = Ux C n = λx C n = λ x C n, hence λ = 1, as desired. Problem 5 Show that the permutation matrices in M n are orthogonal and that the permutation matrices form a subgroup of the group of real orthogonal matrices. How many different permutation matrices are there in M n? Solution. By definition, a matrix P M n is called a permutation matrix if exactly one entry in each row and column is equal to 1, and all other entries are 0. That is, letting e i C n denote the standard basis element of C n that has a 1 in the i th row and zeros elsewhere, and S n be the set of all permutations on n elements, then P = [e σ(1) e σ(n) ] = P σ for some permutation σ S n such that σ(k) denotes the k th member of σ. Observe that for any σ S n, and as { 1 if i = j e T σ(i)e σ(j) = 0 otherwise for 1 i j n by the definition of e i, we have that e T σ(1) Pσ T e σ(1) e T σ(1) e σ(n) P σ =.. = I n (= P σ Pσ T ) e T σ(n) e σ(1) e T σ(n) e σ(n) (where I n denotes the n n identity matrix). Hence Pσ 1 = Pσ T (permutation matrices are trivially nonsingular), and so P σ is (real) orthogonal. Since the above holds for any σ S n, it follows that any permutation matrix is orthogonal. Now, notice that I n is a permutation matrix corresponding to the identity in the group S n, so the set of all permutation matrices in M n is (trivially) nonempty, and contains the identity element of GL n. Moreover, by the preceding paragraph, for each σ S n and each corresponding permutation matrix P σ, Pσ T = Pσ 1, and observe further that Pσ 1 = P σ 1, since P σ has a 1 in column i, row σ(i), and Pσ 1 = Pσ T = P τ has a 1 in column σ(i), row τ(σ(i)) = i for all i = 1,..., n. Thus τ σ = e, the identity element of S n, so τ = σ 1 since S n is a group. As such, the inverse (transpose) of a permutation matrix is again a permutation matrix. Finally, if ν S n is any other permutation, then the preceding discussion 1

2 shows that e T σ(1) Pσ T e ν(1) e T σ(1) e ν(n) P ν =.. = [e (σ ν)(1) e (σ ν)(n) ] = P σ ν e T σ(n) e ν(1) e T σ(n) e ν(n) hence as σ ν S n, the product of permutation matrices is again a permutation matrix (this is rather trivial given the definition of permutation matrix, but it illustrates the connection between permutation matrices in M n and permutations in S n ). Therefore, the set of all permutation matrices is not only a subgroup of GL n, but since each is orthogonal, such is a subgroup of the set of all orthogonal matrices as well. Moreover, the mapping σ P σ is a bijection, hence as o(s n ) = n!, it follows that there are n! different permutation matrices in M n (thus, the order of the subgroup in question is n!). Problem 9 If U M n is unitary, show that U, U T, and U are all unitary. Solution. Let U M n be unitary. That U is unitary follows readily from Theorem 2.1.4(d); that U T is unitary follows from the fact that as the columns of U form an orthonormal set by Theorem 2.1.4(e), then the rows of U T form an orthonormal set. Now, since U = U T is unitary, the rows of U T form an orthonormal set, hence the columns of U form an orthonormal set, and thus U is unitary. Problem 12 Show that if A M n is similar to a unitary matrix, then A 1 is similar to A. Solution. If A M n is similar to the unitary matrix U, then there is a nonsingular matrix S such that U = SAS 1, hence AS 1 = S 1 U, and as such A(S 1 U S) = S 1 UU S = S 1 S = I n. Since S and U are nonsingular, S 1 U S is nonsingular, hence it follows that A is nonsingular (by the exercise preceding Theorem 2.1.4). Thus A 1 = S 1 U S, so that U = SA 1 S 1, and so as U = SAS 1, it follows that U = (S 1 ) A S = SA 1 S 1, and therefore, since S S is nonsingular and (S 1 ) = (S ) 1 by the non-singularity of S, A 1 = S 1 (S 1 ) A S S = (S S) 1 A S S, which implies that A 1 and A are similar. Section 2.3: Schur s Unitary Triangularization Theorem Problem 3 Let A M n (R). Explain why the nonreal eigenvalues of A (if any) must occur in conjugate pairs. Solution. A simple answer to the given question is that since A M n (R), the characteristic polynomial p A (t) has real coefficients, and hence any nonreal roots occur in conjugate pairs, it follows that any nonreal eigenvalues of A must occur in conjugate pairs. This also follows by Theorem 2.3.4, since there is a real orthogonal matrix Q M n (R) such that Q T AQ M n (R) where Q T AQ = A 1 A 2 0 A k, 2

3 and each A i is a real 1 1 matrix (so A i σ(a)), or a real 2 2 matrix with a nonreal pair of complex conjugate eigenvalues. Hence, since σ(a) = σ(q T AQ) by similarity, any nonreal eigenvalues of A must occur in conjugate pairs. Problem 6 Let A, B M n be given, and suppose A and B are simultaneously similar to upper triangular matrices; that is, S 1 AS and S 1 BS are both upper triangular for some nonsingular S M n. Show that every eigenvalue of AB BA must be zero. Solution. Put T A = S 1 AS and T B = S 1 BS. Since T A and T B are upper triangular, T A T B and T B T A are upper triangular, hence as T A T B = S 1 ASS 1 BS = S 1 ABS and similarly T B T A = S 1 BAS. Now, T A T B T B T A = S 1 ABS S 1 BAS = S 1 (AB BA)S, hence as T A T B and T B T A are both upper triangular, it follows further that T A T B T B T A is also upper triangular, hence the eigenvalues of AB BA are the diagonal elements of T A T B T B T A. But, if T A = [t ij ], T B = [s ij ], then t ij = s ij = 0 if i > j, hence it follows that T A T B = t 11 * 0 t nn s 11 * 0 s nn t 11 s 11 * 0 t nn s nn so the diagonal of T A T B is t ii s ii, i = 1,..., n, and by a similar computation, the diagonal of T B T A is s ii t ii (i.e. the two set of diagonal entries are the same). Therefore, the diagonal of T A T B T B T A is t ii s ii s ii t ii = 0 for all i = 1,..., n, which implies that every eigenvalue of AB BA is zero, as desired. Section 2.4: Some Implications of Schur s Theorem Problem 2 If A M n, show that the rank of A is not less than the number of nonzero eigenvalues of A. Solution. If A M n and σ(a) = {λ 1,..., λ n }, then by Schur s Theorem, there is a unitary matrix U such that U AU = T = [t ij ] where T is upper triangular and t ii = λ i, i = 1,..., n. If k of the eigenvalues of A are nonzero, then T has k nonzero and n k zero entries along its main diagonal. As such, the k columns containing the nonzero eigenvalues of A constitute a linearly independent set (since T is upper triangular), and as such, rank(t ) k. But then rank(a) k since U is nonsingular and rank is invariant under multiplication by nonsingular matrices. Of course, we may certainly have rank(a) > k, for if A = [ 0 1 then A is already upper triangular, and σ(a) = {0}, so even though A has no nonzero eigenvalues, rank(a) = 1 > 0. Problem 4 Let A M n be a nonsingular matrix. Show that any matrix that commutes with A also commutes with A 1. ],, 3

4 Solution. Here, we provide two proofs of the given statement. First, if A M n is nonsingular and AB = BA for some B M n, then B = A 1 BA hence BA 1 = A 1 B, so B commutes with A 1 if and only if it commutes with A. Second, by Corollary 2.4.4, since A M n is nonsingular, there is a polynomial q(t), whose coefficients depend on A and where deg(q(t)) n 1, such that A 1 = q(a). Put k = deg(q(t)) and write q(t) = a k t k + a k 1 t k a 1 t + a 0, where a k 0. Now, observe that showing BA = AB implies that Bq(A) = q(a)b will prove the given statement. Note that for any p N, we have BA p = BAA p 1 = ABA p 1 = = A i BA p i = = A p 1 BA = A p B, so B commutes with any positive integer power of A; as such, we compute and thus A 1 B = BA 1, as desired. Problem 5 q(a)b = (a k A k + a k 1 A k a 1 A + a 0 )B = a k A k B + a k 1 A k 1 B + + a 1 AB + a 0 IB = a k BA k + a k 1 BA k a 1 BA + a 0 BI = B(a k A k + a k 1 A k a 1 A + a 0 I) = Bq(A), Use (2.3.1) to show that if A M n has eigenvalues λ 1, λ 2,..., λ n, then n λ k i = tr(a k ), k = 1, 2,... Solution. First, if A M n, and σ(a) = {λ 1,..., λ n }, then letting p(t) = t k for k = 1, 2,..., by Theorem we have that p(a) = A k has eigenvalues p(λ i ) = λ k i, i = 1,..., n. Now, by Schur s Theorem, for each k = 1, 2,..., there is a unitary matrix U k M n such that Uk Ak U k = T k = [t (k) ij ] where T k is upper triangular, and t (k) ii = λ k i, i = 1,..., n. Hence, by Problem 11 (below), as tr(ab) = tr(ba), and as the trace of a matrix is the sum of the eigenvalues of the matrix that tr(a k ) = tr(u k U k A k ) = tr(u k A k U k ) = tr(t k ) = n λ k i k = 1, 2,... as desired. Problem 9 Let A M n, B M m be given and suppose A and B have no eigenvalues in common; that is, σ(a) σ(b) is empty. Use the Cayley-Hamilton theorem (2.4.2) to show that the equation AX XB = 0, X M n,m has only the solution X = 0. Deduce from this fact that the equation AX XB = C has a unique solution X M n,m for each given C M n,m. Solution. Suppose AX = XB, for A, B, and X as given above. Then, assuming that A k X = XB k for k = 1,..., p, we have A p+1 X = A(A p X) = A(XB p ) = (AX)B p = (XB)B p = XB p+1, thus by induction, A k X = XB k for all k = 1, 2,.... In this way, if p(t) is any polynomial, it follows 4

5 that p(a)x = Xp(B) (as in Problem 4 above). So p A (A)X = Xp A (B), hence as p A (A) = 0 by the Cayley-Hamilton Theorem, we have Xp A (B) = 0. But, since p A (t) = (t λ 1 )(t λ 2 ) (t λ n ) where λ i σ(a), i = 1,..., n, it follows that p A (B) = n (B λ i I). Moreover, the eigenvalues of the matrix p A (B) are p A (µ j ) for µ j σ(a) σ(b) =, µ j λ i for any 1 i n and 1 j m, so σ(b), j = 1,..., m, hence as p A (µ j ) = n (µ j λ i ) 0 for each j = 1,..., m. So, as all of the eigenvalues of p A (B) are nonzero, it follows that p A (B) is nonsingular, and as such, Xp A (B) = 0 has the unique solution X = 0, hence AX XB = 0 has the unique solution X = 0. So, considering the linear transformation T : M n,m M n,m where T (X) = AX XB, as T (X) = 0 has the unique solution X = 0, it follows that T (X) = C has a unique solution for each C M n,m, and the proof is complete. Problem 11 Let A, B M n be given and consider the commutator C = AB BA. Show that tr(c) = 0. Consider [ ] [ ] 0 1 A = and B = 1 0 and show that a commutator need not be nilpotent; that is, some eigenvalues of a commutator can be nonzero, even though the sum of the eigenvalues must be zero. Solution. First, by the definition of trace as the sum of diagonal entries, we have tr(c) = tr(ab BA) = tr(ab) tr(ba), hence by Theorem , as the eigenvalues of AB and BA are the same (counting multiplicity), and as the trace of a matrix is also the sum of its eigenvalues, we have that tr(ab) = tr(ba), so that tr(c) = 0. Now, observe that with A and B as given above, we have [ ] 1 0 C = AB BA = 0 1 so that C has (nonzero) eigenvalues 1 and 1, and hence C is not nilpotent, but we see that tr(c) = = 0. 5

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

MATRICES ARE SIMILAR TO TRIANGULAR MATRICES

MATRICES 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 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

Math 489AB Exercises for Chapter 1 Fall Section 1.0

Math 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 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

1 Linear Algebra Problems

1 Linear Algebra Problems Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and

More information

Problems in Linear Algebra and Representation Theory

Problems 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 information

Chapter 1. Matrix Calculus

Chapter 1. Matrix Calculus Chapter 1 Matrix Calculus 11 Definitions and Notation We assume that the reader is familiar with some basic terms in linear algebra such as vector spaces, linearly dependent vectors, matrix addition and

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

Lecture 23: Trace and determinants! (1) (Final lecture)

Lecture 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 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

Foundations of Matrix Analysis

Foundations of Matrix Analysis 1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the

More information

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

Linear algebra I Homework #1 due Thursday, Oct Show that the diagonals of a square are orthogonal to one another.

Linear algebra I Homework #1 due Thursday, Oct Show that the diagonals of a square are orthogonal to one another. Homework # due Thursday, Oct. 0. Show that the diagonals of a square are orthogonal to one another. Hint: Place the vertices of the square along the axes and then introduce coordinates. 2. Find the equation

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

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

Numerical Linear Algebra Homework Assignment - Week 2

Numerical 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 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

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88 Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

More information

Math 108b: Notes on the Spectral Theorem

Math 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 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

MATH 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. 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 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

Chap 3. Linear Algebra

Chap 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 information

GRE 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. 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 information

MAT 2037 LINEAR ALGEBRA I web:

MAT 2037 LINEAR ALGEBRA I web: MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear

More information

Exercise Sheet 1.

Exercise Sheet 1. Exercise Sheet 1 You can download my lecture and exercise sheets at the address http://sami.hust.edu.vn/giang-vien/?name=huynt 1) Let A, B be sets. What does the statement "A is not a subset of B " mean?

More information

Matrices A brief introduction

Matrices A brief introduction Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 41 Definitions Definition A matrix is a set of N real or complex

More information

Example: Filter output power. maximization. Definition. Eigenvalues, eigenvectors and similarity. Example: Stability of linear systems.

Example: Filter output power. maximization. Definition. Eigenvalues, eigenvectors and similarity. Example: Stability of linear systems. Lecture 2: Eigenvalues, eigenvectors and similarity The single most important concept in matrix theory. German word eigen means proper or characteristic. KTH Signal Processing 1 Magnus Jansson/Emil Björnson

More information

Math 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 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 information

Linear Algebra Formulas. Ben Lee

Linear Algebra Formulas. Ben Lee Linear Algebra Formulas Ben Lee January 27, 2016 Definitions and Terms Diagonal: Diagonal of matrix A is a collection of entries A ij where i = j. Diagonal Matrix: A matrix (usually square), where entries

More information

1. 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

1. 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 information

NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques

More information

7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4

7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4 7 Matrix Operations Copyright 017, Gregory G. Smith 9 October 017 The product of two matrices is a sophisticated operations with a wide range of applications. In this chapter, we defined this binary operation,

More information

Lecture notes: Applied linear algebra Part 2. Version 1

Lecture notes: Applied linear algebra Part 2. Version 1 Lecture notes: Applied linear algebra Part 2. Version 1 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 First, some exercises: xercise 0.1 (2 Points) Another least

More information

DIAGONALIZATION BY SIMILARITY TRANSFORMATIONS

DIAGONALIZATION BY SIMILARITY TRANSFORMATIONS DIAGONALIZATION BY SIMILARITY TRANSFORMATIONS The correct choice of a coordinate system (or basis) often can simplify the form of an equation or the analysis of a particular problem. For example, consider

More information

Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,

Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, 2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for

More information

MATHEMATICS 217 NOTES

MATHEMATICS 217 NOTES MATHEMATICS 27 NOTES PART I THE JORDAN CANONICAL FORM The characteristic polynomial of an n n matrix A is the polynomial χ A (λ) = det(λi A), a monic polynomial of degree n; a monic polynomial in the variable

More information

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

More information

LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS

LINEAR 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 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

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

MATH 235. Final ANSWERS May 5, 2015

MATH 235. Final ANSWERS May 5, 2015 MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

More information

Linear Algebra. Workbook

Linear Algebra. Workbook Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

More information

Matrices and Matrix Algebra.

Matrices and Matrix Algebra. Matrices and Matrix Algebra 3.1. Operations on Matrices Matrix Notation and Terminology Matrix: a rectangular array of numbers, called entries. A matrix with m rows and n columns m n A n n matrix : a square

More information

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T. Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where

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

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

Massachusetts Institute of Technology Department of Economics Statistics. Lecture Notes on Matrix Algebra

Massachusetts Institute of Technology Department of Economics Statistics. Lecture Notes on Matrix Algebra Massachusetts Institute of Technology Department of Economics 14.381 Statistics Guido Kuersteiner Lecture Notes on Matrix Algebra These lecture notes summarize some basic results on matrix algebra used

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: 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 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

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)

More information

Linear Systems and Matrices

Linear 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 information

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

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

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex

More information

1 Last time: least-squares problems

1 Last time: least-squares problems MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

More information

FALL 2011, SOLUTION SET 10 LAST REVISION: NOV 27, 9:45 AM. (T c f)(x) = f(x c).

FALL 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 information

Eigenvalue and Eigenvector Problems

Eigenvalue and Eigenvector Problems Eigenvalue and Eigenvector Problems An attempt to introduce eigenproblems Radu Trîmbiţaş Babeş-Bolyai University April 8, 2009 Radu Trîmbiţaş ( Babeş-Bolyai University) Eigenvalue and Eigenvector Problems

More information

Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

More information

MTH 102: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur. Problem Set

MTH 102: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur. Problem Set MTH 102: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set 6 Problems marked (T) are for discussions in Tutorial sessions. 1. Find the eigenvalues

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe ECE133A (Winter 2018) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane. Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 8 Lecture 8 8.1 Matrices July 22, 2018 We shall study

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

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

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

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

Bare-bones outline of eigenvalue theory and the Jordan canonical form

Bare-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 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

MAT Linear Algebra Collection of sample exams

MAT 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 information

The Singular Value Decomposition

The Singular Value Decomposition The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will

More information

MATH 431: FIRST MIDTERM. Thursday, October 3, 2013.

MATH 431: FIRST MIDTERM. Thursday, October 3, 2013. MATH 431: FIRST MIDTERM Thursday, October 3, 213. (1) An inner product on the space of matrices. Let V be the vector space of 2 2 real matrices (that is, the algebra Mat 2 (R), but without the mulitiplicative

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.

The 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 information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 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 information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX 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 information

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

More information

Topic 1: Matrix diagonalization

Topic 1: Matrix diagonalization Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it

More information

Math 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 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 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

G1110 & 852G1 Numerical Linear Algebra

G1110 & 852G1 Numerical Linear Algebra The University of Sussex Department of Mathematics G & 85G Numerical Linear Algebra Lecture Notes Autumn Term Kerstin Hesse (w aw S w a w w (w aw H(wa = (w aw + w Figure : Geometric explanation of the

More information

Chapter 4. Matrices and Matrix Rings

Chapter 4. Matrices and Matrix Rings Chapter 4 Matrices and Matrix Rings We first consider matrices in full generality, i.e., over an arbitrary ring R. However, after the first few pages, it will be assumed that R is commutative. The topics,

More information

Matrices A brief introduction

Matrices A brief introduction Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 44 Definitions Definition A matrix is a set of N real or complex

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

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors /88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix

More information

Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions

Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall 2015 Midterm Exam Review Solutions Practice exam questions: 1. Let V 1 R 2 be the subset of all vectors whose slope

More information

Homework 1 Elena Davidson (B) (C) (D) (E) (F) (G) (H) (I)

Homework 1 Elena Davidson (B) (C) (D) (E) (F) (G) (H) (I) CS 106 Spring 2004 Homework 1 Elena Davidson 8 April 2004 Problem 1.1 Let B be a 4 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange

More information

Lecture 15 Review of Matrix Theory III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 15 Review of Matrix Theory III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 15 Review of Matrix Theory III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Matrix An m n matrix is a rectangular or square array of

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

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

E2 212: Matrix Theory (Fall 2010) Solutions to Test - 1

E2 212: Matrix Theory (Fall 2010) Solutions to Test - 1 E2 212: Matrix Theory (Fall 2010) s to Test - 1 1. Let X = [x 1, x 2,..., x n ] R m n be a tall matrix. Let S R(X), and let P be an orthogonal projector onto S. (a) If X is full rank, show that P can be

More information

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2 Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an

More information

Notes on Eigenvalues, Singular Values and QR

Notes on Eigenvalues, Singular Values and QR Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square

More information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

j=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p.

j=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p. LINEAR ALGEBRA Fall 203 The final exam Almost all of the problems solved Exercise Let (V, ) be a normed vector space. Prove x y x y for all x, y V. Everybody knows how to do this! Exercise 2 If V is a

More information

Math 102 Final Exam - Dec 14 - PCYNH pm Fall Name Student No. Section A0

Math 102 Final Exam - Dec 14 - PCYNH pm Fall Name Student No. Section A0 Math 12 Final Exam - Dec 14 - PCYNH 122-6pm Fall 212 Name Student No. Section A No aids allowed. Answer all questions on test paper. Total Marks: 4 8 questions (plus a 9th bonus question), 5 points per

More information

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September

More information

CHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then

CHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then LECTURE 6: POSITIVE DEFINITE MATRICES Definition: A Hermitian matrix A Mn is positive definite (pd) if x Ax > 0 x C n,x 0 A is positive semidefinite (psd) if x Ax 0. Definition: A Mn is negative (semi)definite

More information