Linear Algebra: Characteristic Value Problem

Size: px
Start display at page:

Download "Linear Algebra: Characteristic Value Problem"

Transcription

1 Linear Algebra: Characteristic Value Problem

2 . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number 2 { and a non-zero vector x 2 { n such that Ax = x? () This is known as the characteristic value problem or the eigenvalue problem. If x 6= 0 and satisfy the equation Ax = x; then is called a characteristic value or eigenvalue of A; and x is called a characteristic vector or eigenvector of A: Clearly, () holds if and only if (A I) x = 0: (2) - But (2) holds for non-zero x if and only if the column vectors of (A I) are linearly dependent, that is, ja Ij = 0: (3)

3 2 This equation is called the characteristic equation of A: Consider the expression f () = ja Ij : (4) - f is a polynomial of degree n in : It is called the characteristic polynomial of A. Example: Consider the 2 2 matrix A = 2 2 : The characteristic equation is: 2 2 = 0; ) (2 )2 = 0; ) ( ) ( 3) = 0: Thus, the characteristic roots are = and = 3:

4 3 Putting = in (2), we get which yields x x 2 = x + x 2 = 0: - Thus the general solution of the characteristic vector corresponding to the characteristic root = is given by 0 0 (x ; x 2 ) = (; ) ; for 6= 0: - Similarly, corresponding to the characteristic root = 3; we have the characteristic vector given by (x ; x 2 ) = (; ) ; for 6= 0:

5 In general, the characteristic equation will have n roots in the complex plane (by the Fundamental Theorem of Algebra ), since it is a polynomial equation (in ) of degree n: (Of course some of these roots might be repeated.) In general, the corresponding eigenvectors will also have their components in the complex plane. 4 Example: Consider the 2 2 matrix A = 2 2 : Check that the eigenvalues are complex numbers and the eigenvectors have complex components.

6 2. Characteristic Values, Trace & Determinant of a Matrix If A is an n n matrix, the trace of A; denoted by tr(a), is the number dened by nx tr(a) = a ii : a a #. Consider the 2 2 matrix A = 2 : a 2 a 22 Write down the characteristic equation. Suppose and 2 are two characteristic values of A: Prove that (a) + 2 = tr(a); and (b) 2 = jaj : In general, for an n n matrix A; with eigenvalues ; 2 ; :::; n ; we have nx ny i = tr(a); and i = jaj : i= i= i= 5

7 3. Characteristic Values & Vectors of Symmetric Matrices There is considerable simplication in the theory of characteristic values if A is a symmetric matrix. Theorem : If A is an n n symmetric matrix, then all the eigenvalues of A are real numbers and its eigenvectors are real vectors. We will develop the theory of eigenvalues and eigenvectors only for symmetric matrices. Normalized Eigenvectors: Note that if x is an eigenvector corresponding to an eigenvalue ; then so is tx; where t is any non-zero scalar. So we normalize the eigenvectors. A normalized eigenvector is an eigenvector with (Euclidean) norm equal to. 6

8 7 Let x = (x ; x 2 ; :::; x n ) be an eigenvector with norm kxk = x 2 + x ::: + x 2 n - Since x is a non-zero vector, kxk > 0: Dene x y = kxk ; x 2 kxk ; :::; x n : kxk Now we have kyk = ; that is, y is a normalized eigenvector. 2 :

9 8 4. Spectral Decomposition of Symmetric Matrices (To proceed, we specialize further we consider symmetric matrices with distinct eigenvalues, that is, i 6= j :) Orthogonal Matrix: An nn matrix C is called an orthogonal matrix if C is invertible and its inverse equals its transpose, that is, C T = C : Theorem 2: Let A be an n n symmetric matrix with n distinct eigenvalues, ; 2 ; :::; n : If y ; y 2 ; :::; y n are (normalized) eigenvectors corresponding to the eigenvalues ; 2 ; :::; n ; then the matrix B such that B i ; the i-th column vector of B; is the vector y i (i = ; 2; :::; n), is an orthogonal matrix. Proof: To be discussed in class.

10 9 Hints: 3 steps: - Step. If i 6= j; then y i T y j = 0: Use the denition (equation ()) to prove that same time, y j T Ay i = j y j T y i : y j T Ay i = i y j T y i and, at the - Step 2. B is invertible. Show that the vectors y ; y 2 ; :::; y n are linearly independent. - Step 3. B = B T. Show directly that B T B = I (the identity matrix).

11 Theorem 3 (Spectral Decomposition): Let A be an n n symmetric matrix with n distinct eigenvalues, ; 2 ; :::; n : Let y ; y 2 ; :::; y n are (normalized) eigenvectors corresponding to the eigenvalues ; 2 ; :::; n : Let B be the n n matrix such that B i ; the i-th column vector of B; is the vector y i (i = ; 2; :::; n). Then A = BLB T where L is the diagonal matrix with the eigenvalues of A ( ; 2 ; :::; n ) on its diagonal. Remark: The above expression shows that matrix A can be decomposed into a matrix L consisting of its eigenvalues on the diagonal and the matrices B and B T which consists of its eigenvectors. Proof: To be discussed in class. 0

12 Hints: Consider any two n n matrices, C and D: - Let (C ; C 2 ; :::; C n ) be the set of row vectors of C and D ; D 2 ; :::; D n be the set of column vectors of D: - To prove Theorem 3 use the following observation on matrix multiplication: 0 C CD = B C 2 A D D 2 D n C 4 = 0 C D C D 2 C D n B C 2 D C 2 D 2 C 2 D n A C n D C n D 2 C n D n = CD CD 2 CD n :

13 5. Quadratic Forms 2 A quadratic form on < n is a real-valued function of the form nx nx Q (x ; x 2 ; :::; x n ) = a ij x i x j i= j= = a x x + a 2 x x 2 + ::: + a n x x n +a 2 x 2 x + a 22 x 2 x 2 + ::: + a 2n x 2 x n +::: + a n x n x + a n2 x n x 2 + ::: + a nn x n x n : A quadratic form Q can be represented by a matrix A so that Q (x) = x T Ax where A nn = 0 a a 2 a n a 2. a 22. a 2n.... a n a n2 a nn C A and x n = 0 x x 2. x n C A :

14 3 Note that a ij and a ji are both coefcients of x i x j when i 6= j: The coefcient of x i x j is (a ij + a ji ) when i 6= j: If a ij 6= a ji ; we can uniquely dene new coefcients b ij = b ji = a ij + a ji ; for all i; j 2 so that b ij + b ji = a ij + a ji ; and B = (b ij ) = B T ; that is B is a symmetric matrix. This redenition of the coefcients does not change the value of Q for any x: Thus we can always assume that the matrix A associated with the quadratic form x T Ax is symmetric.

15 4 Examples: The general quadratic form in two variables is a x 2 + 2a 2 x x 2 + a 22 x 2 2: In matrix form this can be written as a a (x x 2 ) 2 a 2 a 22 x x 2 : The general quadratic form in three variables a x 2 + a 22 x a 33 x a 2 x x 2 + a 3 x x 3 + a 23 x 2 x 3 can be written as 0 (x x 2 x 3 ) a 2 a 2 2 a 3 2 a 2 a 22 2 a 23 2 a 3 2 a 23 a 33 0 C B x x 2 x 3 C A :

16 5 Deniteness of Quadratic Forms: Let A be a symmetric n n matrix. Then A is positive denite if x T Ax > 0 for all x in < n ; x 6= 0: A is negative denite if x T Ax < 0 for all x in < n ; x 6= 0: A is positive semi-denite if x T Ax 0 for all x in < n : A is negative semi-denite if x T Ax 0 for all x in < n : A is indenite if x T Ax > 0 for some x in < n and x T Ax < 0 for some other x in < n : Remarks: Consider, for example, the denition of positive deniteness. Note that the relevant inequality must hold for every vector x 6= 0 in < n : - Observation : If we know that A is positive denite, then we should be able to infer some useful properties of A quite easily.

17 #2. Prove that if a symmetric n n matrix A is positive denite then all its diagonal elements must be positive. - Observation 2: On the other hand, if we do not know that A is positive denite, then the above denition by itself will not be very easy to check to determine whether A is positive denite or not.! This observation leads one to explore convenient characterizations of quadratic forms. 6

18 6. Characterization of Quadratic Forms 7 Theorem 4: Let A be a symmetric n n matrix with distinct eigenvalues, ; 2 ; :::; n : Then (a) A is positive (negative) denite if and only if every eigenvalue of A is positive (negative); (b) A is positive (negative) semi-denite if and only if every eigenvalue of A is nonnegative (non-positive); (c) A is indenite if and only if A has a positive eigenvalue and a negative eigenvalue. Proof: To be discussed in class. Hints: Use the spectral decomposition of A given in Theorem 3. #3. Examples: Consider the following matrices: 0 A = ; B = ; C = Find out their eigenvalues, and characterize them (in terms of deniteness).

19 7. Alternative Characterization of Quadratic Forms 8 An alternative way to characterize quadratic forms is in terms of the signs of the principal minors of the corresponding matrix. If A is an n n matrix, a principal minor of order r is the determinant of the r r submatrix that remains when (n r) rows and (n r) columns with the same indices are deleted from A: Example: Consider the 3 3 matrix 0 A a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 A : The principal minors of order 2 are: a a 2 a 2 a 22 ; a a 3 a 3 a 33 ; a 22 a 23 a 32 a 33 :

20 9 The principal minors of order are: The principal minors of order 3 is: jaj : a ; a 22 ; a 33 : If A is an n n matrix, the leading principal minor of order r is dened as a a r A r =..... a r a rr : Example: For the 3 3 matrix A, the three leading principal minors are A = a ; A 2 = a a 2 a 2 a 22 ; A 3 = jaj :

21 20 Theorem 5: Let A be a symmetric n n matrix. Then (a) A is positive denite if and only if all its n leading principal minors are positive. (b) A is negative denite if and only if all its n leading principal minors alternate in sign, starting with negative. (That is, the r-th leading principal minor, A r ; r = ; 2; :::; n; has the same sign as ( ) r :) (c) If some r-th leading principal minor of A (or some pair of them) is non-zero but does not t into either of the two sign patterns in (a) and (b), then A is indenite. (d) A is positive semi-denite if and only if every principal minor of A of every order is non-negative. (e) A is negative semi-denite if and only if every principal minor of A of odd order is non-positive and every principal minor of even order is non-negative. Proof: See section 6.4 (pages ) of the textbook.

22 2 Examples: Consider the matrices studied above: 0 A = ; B = ; C = #4. Characterize the matrices A; B and C (in terms of deniteness) using the principal minors criteria.

23 22 References Must read the following sections from the textbook: Section 23. (pages ): Denitions and Examples of Eigenvalues and Eigenvectors; Section 23.3 (pages ): Properties of Eigenvalues; Section 23.7 (pages ): Symmetric Matrices; Sections 6. and 6.2 (pages ): Quadratic Forms and Deniteness of Quadratic Forms; Section 23.8 (pages ): Deniteness of Quadratic Forms. Most of the materials covered can be found in. Hohn, Franz E., Elementary Matrix Algebra, New Delhi: Amerind, 97 (chapters 9, 0), 2. Hadley, G., Linear Algebra, Massachusetts: Addison-Wesley, 964 (chapter 7).

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

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

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

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

Linear Algebra. Shan-Hung Wu. Department of Computer Science, National Tsing Hua University, Taiwan. Large-Scale ML, Fall 2016

Linear Algebra. Shan-Hung Wu. Department of Computer Science, National Tsing Hua University, Taiwan. Large-Scale ML, Fall 2016 Linear Algebra Shan-Hung Wu shwu@cs.nthu.edu.tw Department of Computer Science, National Tsing Hua University, Taiwan Large-Scale ML, Fall 2016 Shan-Hung Wu (CS, NTHU) Linear Algebra Large-Scale ML, Fall

More information

Matrix Algebra: Summary

Matrix Algebra: Summary May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................

More information

[3] (b) Find a reduced row-echelon matrix row-equivalent to ,1 2 2

[3] (b) Find a reduced row-echelon matrix row-equivalent to ,1 2 2 MATH Key for sample nal exam, August 998 []. (a) Dene the term \reduced row-echelon matrix". A matrix is reduced row-echelon if the following conditions are satised. every zero row lies below every nonzero

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

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

MATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y.

MATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y. as Basics Vectors MATRIX ALGEBRA An array of n real numbers x, x,, x n is called a vector and it is written x = x x n or x = x,, x n R n prime operation=transposing a column to a row Basic vector operations

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

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

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

18.S34 linear algebra problems (2007)

18.S34 linear algebra problems (2007) 18.S34 linear algebra problems (2007) Useful ideas for evaluating determinants 1. Row reduction, expanding by minors, or combinations thereof; sometimes these are useful in combination with an induction

More information

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

More information

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

More information

Math Matrix Algebra

Math Matrix Algebra Math 44 - Matrix Algebra Review notes - (Alberto Bressan, Spring 7) sec: Orthogonal diagonalization of symmetric matrices When we seek to diagonalize a general n n matrix A, two difficulties may arise:

More information

Math 308 Practice Final Exam Page and vector y =

Math 308 Practice Final Exam Page and vector y = Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

Rank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col

Rank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col Review of Linear Algebra { E18 Hanout Vectors an Their Inner Proucts Let X an Y be two vectors: an Their inner prouct is ene as X =[x1; ;x n ] T Y =[y1; ;y n ] T (X; Y ) = X T Y = x k y k k=1 where T an

More information

Review of Vectors and Matrices

Review of Vectors and Matrices A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P

More information

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 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 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

More information

Large Scale Data Analysis Using Deep Learning

Large Scale Data Analysis Using Deep Learning Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset

More information

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12 24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2

More information

Principal Components Theory Notes

Principal Components Theory Notes Principal Components Theory Notes Charles J. Geyer August 29, 2007 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the University of Minnesota, Spring 2006. This not a theory

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Matrix Algebra, part 2

Matrix Algebra, part 2 Matrix Algebra, part 2 Ming-Ching Luoh 2005.9.12 1 / 38 Diagonalization and Spectral Decomposition of a Matrix Optimization 2 / 38 Diagonalization and Spectral Decomposition of a Matrix Also called Eigenvalues

More information

MTH 5102 Linear Algebra Practice Final Exam April 26, 2016

MTH 5102 Linear Algebra Practice Final Exam April 26, 2016 Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write

More information

Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions

Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Main problem of linear algebra 2: Given

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

Econ Slides from Lecture 8

Econ Slides from Lecture 8 Econ 205 Sobel Econ 205 - Slides from Lecture 8 Joel Sobel September 1, 2010 Computational Facts 1. det AB = det BA = det A det B 2. If D is a diagonal matrix, then det D is equal to the product of its

More information

Symmetric Matrices and Eigendecomposition

Symmetric Matrices and Eigendecomposition Symmetric Matrices and Eigendecomposition Robert M. Freund January, 2014 c 2014 Massachusetts Institute of Technology. All rights reserved. 1 2 1 Symmetric Matrices and Convexity of Quadratic Functions

More information

Mathematical foundations - linear algebra

Mathematical foundations - linear algebra Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar

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

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

11 a 12 a 21 a 11 a 22 a 12 a 21. (C.11) A = The determinant of a product of two matrices is given by AB = A B 1 1 = (C.13) and similarly.

11 a 12 a 21 a 11 a 22 a 12 a 21. (C.11) A = The determinant of a product of two matrices is given by AB = A B 1 1 = (C.13) and similarly. C PROPERTIES OF MATRICES 697 to whether the permutation i 1 i 2 i N is even or odd, respectively Note that I =1 Thus, for a 2 2 matrix, the determinant takes the form A = a 11 a 12 = a a 21 a 11 a 22 a

More information

Review of Some Concepts from Linear Algebra: Part 2

Review of Some Concepts from Linear Algebra: Part 2 Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set

More information

Basic Calculus Review

Basic Calculus Review Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V

More information

Chapter 1. Matrix Algebra

Chapter 1. Matrix Algebra ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface

More information

ENGR-1100 Introduction to Engineering Analysis. Lecture 21. Lecture outline

ENGR-1100 Introduction to Engineering Analysis. Lecture 21. Lecture outline ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility

More information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

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

Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015

Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic

More information

Basic Concepts in Matrix Algebra

Basic Concepts in Matrix Algebra Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1

More information

Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

More information

ENGR-1100 Introduction to Engineering Analysis. Lecture 21

ENGR-1100 Introduction to Engineering Analysis. Lecture 21 ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility

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 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

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

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

More information

3 (Maths) Linear Algebra

3 (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 information

Chapter 3. Determinants and Eigenvalues

Chapter 3. Determinants and Eigenvalues Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

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

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

Mathematical foundations - linear algebra

Mathematical foundations - linear algebra Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar

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

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 Matrices and Systems of Linear Equations. a 1n a 2n March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real

More information

Exercise Set 7.2. Skills

Exercise Set 7.2. Skills Orthogonally diagonalizable matrix Spectral decomposition (or eigenvalue decomposition) Schur decomposition Subdiagonal Upper Hessenburg form Upper Hessenburg decomposition Skills Be able to recognize

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

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

Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.

Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown. Quiz 1) Simplify 9999 999 9999 998 9999 998 2) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown Solving Linear Algebraic Equa3ons Basic Concepts Here only real

More information

Appendix A: Matrices

Appendix A: Matrices Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows

More information

Introduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Introduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional space

More information

Deep Learning Book Notes Chapter 2: Linear Algebra

Deep Learning Book Notes Chapter 2: Linear Algebra Deep Learning Book Notes Chapter 2: Linear Algebra Compiled By: Abhinaba Bala, Dakshit Agrawal, Mohit Jain Section 2.1: Scalars, Vectors, Matrices and Tensors Scalar Single Number Lowercase names in italic

More information

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers. Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

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

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

1 Inner Product and Orthogonality

1 Inner Product and Orthogonality CSCI 4/Fall 6/Vora/GWU/Orthogonality and Norms Inner Product and Orthogonality Definition : The inner product of two vectors x and y, x x x =.., y =. x n y y... y n is denoted x, y : Note that n x, y =

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

Multivariate Statistical Analysis

Multivariate Statistical Analysis Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions

More information

Linear Algebra Solutions 1

Linear Algebra Solutions 1 Math Camp 1 Do the following: Linear Algebra Solutions 1 1. Let A = and B = 3 8 5 A B = 3 5 9 A + B = 9 11 14 4 AB = 69 3 16 BA = 1 4 ( 1 3. Let v = and u = 5 uv = 13 u v = 13 v u = 13 Math Camp 1 ( 7

More information

2. Matrix Algebra and Random Vectors

2. Matrix Algebra and Random Vectors 2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns

More information

Math 1553, Introduction to Linear Algebra

Math 1553, Introduction to Linear Algebra Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level

More information

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

More 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

Chapter 6. Eigenvalues. Josef Leydold Mathematical Methods WS 2018/19 6 Eigenvalues 1 / 45

Chapter 6. Eigenvalues. Josef Leydold Mathematical Methods WS 2018/19 6 Eigenvalues 1 / 45 Chapter 6 Eigenvalues Josef Leydold Mathematical Methods WS 2018/19 6 Eigenvalues 1 / 45 Closed Leontief Model In a closed Leontief input-output-model consumption and production coincide, i.e. V x = x

More information

Linear Algebra Primer

Linear Algebra Primer Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................

More information

Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains

Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems

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

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij

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

Practice Exam. 2x 1 + 4x 2 + 2x 3 = 4 x 1 + 2x 2 + 3x 3 = 1 2x 1 + 3x 2 + 4x 3 = 5

Practice Exam. 2x 1 + 4x 2 + 2x 3 = 4 x 1 + 2x 2 + 3x 3 = 1 2x 1 + 3x 2 + 4x 3 = 5 Practice Exam. Solve the linear system using an augmented matrix. State whether the solution is unique, there are no solutions or whether there are infinitely many solutions. If the solution is unique,

More information

3 a 21 a a 2N. 3 a 21 a a 2M

3 a 21 a a 2N. 3 a 21 a a 2M APPENDIX: MATHEMATICS REVIEW G 12.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers 2 A = 6 4 a 11 a 12... a 1N a 21 a 22... a 2N. 7..... 5 a M1 a M2...

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

Linear Algebra. and

Linear Algebra. and Instructions Please answer the six problems on your own paper. These are essay questions: you should write in complete sentences. 1. Are the two matrices 1 2 2 1 3 5 2 7 and 1 1 1 4 4 2 5 5 2 row equivalent?

More information

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2 EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

More information

MATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018

MATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018 MATH 57: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 18 1 Global and Local Optima Let a function f : S R be defined on a set S R n Definition 1 (minimizers and maximizers) (i) x S

More information

Section 3.9. Matrix Norm

Section 3.9. Matrix Norm 3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix

More information

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name. HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your

More information

Problem 1: Solving a linear equation

Problem 1: Solving a linear equation Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

Linear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form

Linear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form Linear algebra II Homework # solutions. Find the eigenvalues and the eigenvectors of the matrix 4 6 A =. 5 Since tra = 9 and deta = = 8, the characteristic polynomial is f(λ) = λ (tra)λ+deta = λ 9λ+8 =

More information

Lecture Summaries for Linear Algebra M51A

Lecture Summaries for Linear Algebra M51A These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

More information

Tutorials in Optimization. Richard Socher

Tutorials in Optimization. Richard Socher Tutorials in Optimization Richard Socher July 20, 2008 CONTENTS 1 Contents 1 Linear Algebra: Bilinear Form - A Simple Optimization Problem 2 1.1 Definitions........................................ 2 1.2

More information

Information About Ellipses

Information About Ellipses Information About Ellipses David Eberly, Geometric Tools, Redmond WA 9805 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view

More information

Linear Algebra for Machine Learning. Sargur N. Srihari

Linear Algebra for Machine Learning. Sargur N. Srihari Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it

More information

Multivariate Gaussian Distribution. Auxiliary notes for Time Series Analysis SF2943. Spring 2013

Multivariate Gaussian Distribution. Auxiliary notes for Time Series Analysis SF2943. Spring 2013 Multivariate Gaussian Distribution Auxiliary notes for Time Series Analysis SF2943 Spring 203 Timo Koski Department of Mathematics KTH Royal Institute of Technology, Stockholm 2 Chapter Gaussian Vectors.

More information