Control Systems. Linear Algebra topics. L. Lanari

Size: px
Start display at page:

Download "Control Systems. Linear Algebra topics. L. Lanari"

Transcription

1 Control Systems Linear Algebra topics L Lanari

2 outline basic facts about matrices eigenvalues - eigenvectors - characteristic polynomial - algebraic multiplicity eigenvalues invariance under similarity transformation invariance of the eigenspace geometric multiplicity diagonalizable matrix: necessary & sufficient condition diagonalizing similarity transformation a more convenient similarity transformation for complex eigenvalues spectral decomposition not diagonalizable (A defective) case: Jordan blocks Lanari: CS - Linear Algebra

3 matrices - notations and terminology p x q matrix M = 0 m m m q m m m q C A m p m p,q m pq M = {m ij : i =,,p & j =,,q} transpose M T = m 0 ij : m 0 ij = m ji vector (column) row vector scalar square matrix rectangular matrix v : n v T : n M : p p M : p q (p 6= q) Lanari: CS - Linear Algebra 3

4 matrices diagonal matrix block diagonal matrix upper triangular matrix 0 0 upper block triangular matrix m m 0 0 C 0 A 0 0 m n M M 0 A 0 0 M 3 m m m n 0 m m n m nn M M 0 M compact notation: M = diag{m i }, compact notation: M = diag{m i }, C A i =,,p i =,,k similarly: lower triangular strictly lower triangular similarly: lower block triangular strictly lower block triangular Lanari: CS - Linear Algebra 4

5 matrices multiplication by a diagonal matrix a a n n a n a nn 3 3 a a n a n a nn 0 n 7 5 = 7 5 = 3 a a n a n n a nn n 3 a a n n a n a nn n = diag{ i } A A det M M 0 M =det(m ) det(m ) det M 0 0 M =det(m ) det(m ) special case determinant of a diagonal matrix = product of the diagonal terms special case useful for the eigenvalue computation Lanari: CS - Linear Algebra 5

6 matrices square matrix non-singular (invertible), det(m) 6= 0 A = A (AB) = B A M = diag{m i } = M = diag{ m i } A A = AA = I A 0 = I det A =det A T det(mn) = det(m)det(n) for square matrices M and N Lanari: CS - Linear Algebra 6

7 eigenvalues & eigenvectors u A generic transformation Au in general Au and u have different directions u Au particular directions such that Au is parallel to u u Au= u or Au scalar (scaling factor) Lanari: CS - Linear Algebra 7

8 eigenvalues & eigenvectors eigenvalue i eigenvector ui Aui = i ui (A - i I )ui = 0 ui belongs to the nullspace of (A - ii) for non-trivial u i 6=0solution to exist it has to be det(a - ii) = 0 the eigenvalues i are the solutions of the characteristic polynomial (of order n = dim(a)) p A ( )=det( I A) = n + a n n + a n n + + a + a 0 i solution of p A ( )=det( I A) =0 same solutions of det (A - I) = 0 Lanari: CS - Linear Algebra 8

9 eigenvalues & eigenvectors p A ( )=det( I A) = n + a n n + a n n + + a + a 0 polynomial with real coefficients set of the n solutions of p A ( ) = 0 is defined as the spectrum of A: ¾(A) being the coefficients real generic eigenvalue i therefore if ( i R then i C ( u i i R i C real components i ( i, u i complex components and i i ) also is a solution pairs u i define ma( i ) = algebraic multiplicity of eigenvalue i as the multiplicity of the solution = i in p A ( ) = 0 Lanari: CS - Linear Algebra 9

10 eigenvalues & eigenvectors special cases diagonal matrix triangular matrix (upper or lower) 0 m m 0 0 C 0 A 0 0 m n 0 m m m n 0 m m n 0 C A 0 0 m nn eigenvalues = {mi} eigenvalues = elements on the main diagonal eigenvalues = {mii} Lanari: CS - Linear Algebra 0

11 eigenvalues - invariance similarity transformation A T T AT det(t ) 6= 0 same eigenvalues as A eigenvalues are invariant under similarity transformations (proof) def Au i = i u i! u i v T i A = v T i i = i v T i! v T i right eigenvector (column) left eigenvector (row) property v T i u j = ij with ( =0 if i 6= j ± Kronecker ij = if i = j delta Lanari: CS - Linear Algebra

12 eigenspace the eigenspace V i corresponding to the eigenvalue i of A is the vector space V i = {u R n Au = i u} or equivalently V i = Ker(A - ii) reminder: span{v, v,, v k } = vector space generated by all possible linear combinations of the vectors v, v,, v k Ker(M) (kernel or nullspace of M) is the subspace of all vectors st Mv = 0 basis of Ker(M) is a set of linearly independent vectors which spans the whole subspace Ker(M) consequence: eigenvector u i associated to i is not unique example A( u i )= Au i = i u i = i ( u i ) all belong to the same linear subspace Lanari: CS - Linear Algebra

13 geometric multiplicity if i eigenvalue with ma( i ) > it will have one or more linearly independent eigenvectors ui 0 0 ex A =, u 0 =, u 0 = A =, with 6= 0, only u 0 = 0 linearly independent linearly independent note that A and A have the same eigenvalues both with ma( ) = same characteristic polynomial p A ( )=( ) Def geometric multiplicity of i as the dimension of the eigenspace associated to i dim(v i )=mg( i ) mg( i )=dim[ker(a ii)] = n rank(a ii) Lanari: CS - Linear Algebra 3

14 geometric multiplicity useful property ex A = A = 0, (A 0 I)=, (A 0 I)= apple mg( i ) apple ma( i ) apple n mg( )==ma( ) mg( )=< ma( ) ex P 0 0 A projection matrix = = u A u = ma( )==mg( ) = 0 u 3 0 A 0A V V 3 u3 u generic point 45 u Lanari: CS - Linear Algebra 4

15 diagonalization Def An (n x n) matrix A is said to be diagonalizable if there exists an invertible (n x n) matrix T such that TAT - is a diagonal matrix Th An (n x n) matrix A is diagonalizable if and only if it has n linearly independent eigenvectors since the eigenvalues are invariant under similarity transformations if A diagonalizable T AT = = diag{ i },i=,,n the elements on the diagonal of are the eigenvalues of A Lanari: CS - Linear Algebra 5

16 diagonalization we need to find T i ui n linearly independent (by hyp) U = u u u n non-singular Au i = i u i, i =,,n in matrix form A u u u n = u u u n C A 0 0 n AU = U Lanari: CS - Linear Algebra 6

17 diagonalization AU = U being U non-singular, we can define T such that T = U AT = T! A = T T! = T AT therefore the diagonalizing similarity transformation is T st T = U = u u u n alternative Nec & Suff condition for diagonalizability A: (n x n) is diagonalizable if and only if mg( i) = ma( i) for every i distinct eigenvalues (real and/or complex) =) (= A diagonalizable Lanari: CS - Linear Algebra 7

18 diagonalization note that since distinct eigenvalues (real and/or complex) =) (= distinct eigenvalues (real and/or complex) =) A diagonalizable linearly independent eigenvectors i 0 0 i diagonal but not distinct eigenvalues Lanari: CS - Linear Algebra 8

19 diagonalization: complex eigenvalues case complex eigenvalues? ( i, i ) i = i + j i u i = u ai + ju bi eigenvalue eigenvector choices diagonalization T = u i u i! D i = T AT = apple i 0 0 i complex elements or real block x (not diagonalization) T = u ai u bi! M i = T AT = apple i i i i real elements real system representation for complex eigenvalues Lanari: CS - Linear Algebra 9

20 diagonalization simultaneous presence of real and complex eigenvalues If A diagonalizable, there exists a non-singular matrix R such that RAR = diag { r,m r+,m r+3,,m q } with r = diag{,, r} real eigenvalues M i = apple i! i! i i for each complex pair of eigenvalues Lanari: CS - Linear Algebra 0

21 diagonalization example = A = /3 = ± 0j = u = j = +0j u = 4 05+j5 u a = u b = T = u u a u b = TAT = Lanari: CS - Linear Algebra

22 A diagonalizable: spectral decomposition Hyp: A diagonalizable A = U U columns (linearly independent) U = u u u n U = 6 4 v T v T rows U U = I v T i u j = ij, i,j =,,n v T n A = U U = u u nu n v T v T 6 4 v T n spectral form of A A = nx i= i u i v T i or eigen-decomposition Lanari: CS - Linear Algebra

23 A diagonalizable: spectral decomposition A = nx i u i vi T i= column row (n x n) P i = u i v T i is the projection matrix on the invariant subspace generated by ui A = 0 u = 0 u = P v = = v T = v T = 0 0 P = P 0 0 = 0 P v = u P v = u v P v v = u u Lanari: CS - Linear Algebra 3

24 not diagonalizable case since in general apple mg( i ) apple ma( i ) apple n if A not diagonalizable then mg( i ) < ma( i ) in this case A is said to be defective each i will have mg( i) blocks Jk with k =,, mg( i ), each of dimension nk Jordan block of dimension nk J k = 6 4 i i i 0 0 i the knowledge of this dimension is out of scope R n k n k J k i I Null space has dimension Lanari: CS - Linear Algebra 4

25 not diagonalizable case (A defective) generalized eigenvector chain of nk generalized eigenvectors (not part of this course) Jordan canonical form (block diagonal) example unique eigenvalue i of matrix A (n x n) mg( i )=p px ma( i )=n = 9 T : k= n k T AT = J = 6 4J J p with J k R n k n k Jordan block of dim nk Lanari: CS - Linear Algebra 5

26 special cases if ma( i) = then mg( i) = if mg( i) = then only one Jordan block of dimension ma( i) apple 0 = ma( )= mg( )= if i unique eigenvalue of matrix A and if rank( ii - A) = ma( i) - then only one Jordan block of dimension ma( i) consequence of the rank-nullity theorem applied to A - ii A ii : R n! R n dim (R n )=dim(ker(a ii)) + dim (Im(A ii)) n nullity(a - ii) rank(a - ii) Lanari: CS - Linear Algebra 6

27 summary A diagonalizable mg( i) = ma( i) for all i A not diagonalizable mg( i) < ma( i) 9 T st T AT = = diag{ i } nx A = i u i vi T i= 9 T st T AT = diag{j k } block diagonal J k = for real & complex i r = diag{,, r} alternative choice for complex ( i, i*) apple i M i = i 6 4 i spectral form mg( i ) Jordan blocks of the form i 0 0 i 0 0 i i Rn k n k Lanari: CS - Linear Algebra 7

28 vocabulary English eigenvalue/eigenvector characteristic polynomial algebraic/geometric multiplicity similar matrix spectral form Italiano autovalore/autovettore polinomio caratteristico molteplicità algebrica/geometrica matrice simile forma spettrale Lanari: CS - Time response 8

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

Control Systems. Dynamic response in the time domain. L. Lanari

Control Systems. Dynamic response in the time domain. L. Lanari Control Systems Dynamic response in the time domain L. Lanari outline A diagonalizable - real eigenvalues (aperiodic natural modes) - complex conjugate eigenvalues (pseudoperiodic natural modes) - phase

More information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

More 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

Definitions for Quizzes

Definitions for Quizzes Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does

More information

Chapter 2 Subspaces of R n and Their Dimensions

Chapter 2 Subspaces of R n and Their Dimensions Chapter 2 Subspaces of R n and Their Dimensions Vector Space R n. R n Definition.. The vector space R n is a set of all n-tuples (called vectors) x x 2 x =., where x, x 2,, x n are real numbers, together

More information

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

More information

Math Final December 2006 C. Robinson

Math Final December 2006 C. Robinson Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the

More information

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

More information

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that

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

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

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true. 1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III

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

Definition (T -invariant subspace) Example. Example

Definition (T -invariant subspace) Example. Example Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

More information

Linear Algebra Practice Final

Linear Algebra Practice Final . Let (a) First, Linear Algebra Practice Final Summer 3 3 A = 5 3 3 rref([a ) = 5 so if we let x 5 = t, then x 4 = t, x 3 =, x = t, and x = t, so that t t x = t = t t whence ker A = span(,,,, ) and a basis

More information

Lecture 1: Review of linear algebra

Lecture 1: Review of linear algebra Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations

More information

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018 Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

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

Eigenspaces. (c) Find the algebraic multiplicity and the geometric multiplicity for the eigenvaules of A.

Eigenspaces. (c) Find the algebraic multiplicity and the geometric multiplicity for the eigenvaules of A. Eigenspaces 1. (a) Find all eigenvalues and eigenvectors of A = (b) Find the corresponding eigenspaces. [ ] 1 1 1 Definition. If A is an n n matrix and λ is a scalar, the λ-eigenspace of A (usually denoted

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

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

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

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

Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

More information

Control Systems. Internal Stability - LTI systems. L. Lanari

Control Systems. Internal Stability - LTI systems. L. Lanari Control Systems Internal Stability - LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable

More information

ELE/MCE 503 Linear Algebra Facts Fall 2018

ELE/MCE 503 Linear Algebra Facts Fall 2018 ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2

More information

Jordan Normal Form and Singular Decomposition

Jordan Normal Form and Singular Decomposition University of Debrecen Diagonalization and eigenvalues Diagonalization We have seen that if A is an n n square matrix, then A is diagonalizable if and only if for all λ eigenvalues of A we have dim(u λ

More information

Chapter 5 Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

More information

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only non-zero entries are on the diagonal This is equivalent to

More information

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

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

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

MATH 304 Linear Algebra Lecture 34: Review for Test 2.

MATH 304 Linear Algebra Lecture 34: Review for Test 2. MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,

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

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

Math 110 Linear Algebra Midterm 2 Review October 28, 2017

Math 110 Linear Algebra Midterm 2 Review October 28, 2017 Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin

More information

MATH 1553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE

MATH 1553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE MATH 553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE In studying for the final exam, you should FIRST study all tests andquizzeswehave had this semester (solutions can be found on Canvas). Then go

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

Lecture 2: Linear operators

Lecture 2: Linear operators Lecture 2: Linear operators Rajat Mittal IIT Kanpur The mathematical formulation of Quantum computing requires vector spaces and linear operators So, we need to be comfortable with linear algebra to study

More information

Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence

Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the left-hand side of the coefficient matrix of a linear system A x = b which could

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

Chapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors

Chapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there

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

MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS

MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on

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

(f + g)(s) = f(s) + g(s) for f, g V, s S (cf)(s) = cf(s) for c F, f V, s S

(f + g)(s) = f(s) + g(s) for f, g V, s S (cf)(s) = cf(s) for c F, f V, s S 1 Vector spaces 1.1 Definition (Vector space) Let V be a set with a binary operation +, F a field, and (c, v) cv be a mapping from F V into V. Then V is called a vector space over F (or a linear space

More information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show

More information

MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

More information

m We can similarly replace any pair of complex conjugate eigenvalues with 2 2 real blocks. = R

m We can similarly replace any pair of complex conjugate eigenvalues with 2 2 real blocks. = R 1 RODICA D. COSTIN Suppose that some eigenvalues are not real. Then the matrices S and are not real either, and the diagonalization of M must be done in C n. Suppose that we want to work in R n only. Recall

More information

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a). .(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

More information

Math 205, Summer I, Week 4b: Continued. Chapter 5, Section 8

Math 205, Summer I, Week 4b: Continued. Chapter 5, Section 8 Math 205, Summer I, 2016 Week 4b: Continued Chapter 5, Section 8 2 5.8 Diagonalization [reprint, week04: Eigenvalues and Eigenvectors] + diagonaliization 1. 5.8 Eigenspaces, Diagonalization A vector v

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

City Suburbs. : population distribution after m years

City Suburbs. : population distribution after m years Section 5.3 Diagonalization of Matrices Definition Example: stochastic matrix To City Suburbs From City Suburbs.85.03 = A.15.97 City.15.85 Suburbs.97.03 probability matrix of a sample person s residence

More information

The Jordan Normal Form and its Applications

The Jordan Normal Form and its Applications The and its Applications Jeremy IMPACT Brigham Young University A square matrix A is a linear operator on {R, C} n. A is diagonalizable if and only if it has n linearly independent eigenvectors. What happens

More information

Jordan Canonical Form

Jordan Canonical Form Jordan Canonical Form Massoud Malek Jordan normal form or Jordan canonical form (named in honor of Camille Jordan) shows that by changing the basis, a given square matrix M can be transformed into a certain

More information

Math 205, Summer I, Week 4b:

Math 205, Summer I, Week 4b: Math 205, Summer I, 2016 Week 4b: Chapter 5, Sections 6, 7 and 8 (5.5 is NOT on the syllabus) 5.6 Eigenvalues and Eigenvectors 5.7 Eigenspaces, nondefective matrices 5.8 Diagonalization [*** See next slide

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

Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

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

Generalized eigenvector - Wikipedia, the free encyclopedia

Generalized eigenvector - Wikipedia, the free encyclopedia 1 of 30 18/03/2013 20:00 Generalized eigenvector From Wikipedia, the free encyclopedia In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that

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

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

Solutions to practice questions for the final

Solutions to practice questions for the final Math A UC Davis, Winter Prof. Dan Romik Solutions to practice questions for the final. You are given the linear system of equations x + 4x + x 3 + x 4 = 8 x + x + x 3 = 5 x x + x 3 x 4 = x + x + x 4 =

More information

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

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

A Brief Outline of Math 355

A Brief Outline of Math 355 A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting

More information

Math 314H Solutions to Homework # 3

Math 314H Solutions to Homework # 3 Math 34H Solutions to Homework # 3 Complete the exercises from the second maple assignment which can be downloaded from my linear algebra course web page Attach printouts of your work on this problem to

More information

Math 313 Chapter 5 Review

Math 313 Chapter 5 Review Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row

More information

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8. Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:

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

spring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra

spring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM

More information

Diagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Diagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics Diagonalization MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Motivation Today we consider two fundamental questions: Given an n n matrix A, does there exist a basis

More information

TBP MATH33A Review Sheet. November 24, 2018

TBP MATH33A Review Sheet. November 24, 2018 TBP MATH33A Review Sheet November 24, 2018 General Transformation Matrices: Function Scaling by k Orthogonal projection onto line L Implementation If we want to scale I 2 by k, we use the following: [

More information

Lec 2: Mathematical Economics

Lec 2: Mathematical Economics Lec 2: Mathematical Economics to Spectral Theory Sugata Bag Delhi School of Economics 24th August 2012 [SB] (Delhi School of Economics) Introductory Math Econ 24th August 2012 1 / 17 Definition: Eigen

More information

University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm

University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm Name: The proctor will let you read the following conditions

More information

Linear transformations

Linear transformations Linear transformations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Linear transformations

More information

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013 Math Exam Sample Problems Solution Guide October, Note that the following provides a guide to the solutions on the sample problems, but in some cases the complete solution would require more work or justification

More information

Row Space and Column Space of a Matrix

Row Space and Column Space of a Matrix Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them. Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

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 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

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

A proof of the Jordan normal form theorem

A proof of the Jordan normal form theorem A proof of the Jordan normal form theorem Jordan normal form theorem states that any matrix is similar to a blockdiagonal matrix with Jordan blocks on the diagonal. To prove it, we first reformulate it

More information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

235 Final exam review questions

235 Final exam review questions 5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

More 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

Control Systems. Laplace domain analysis

Control Systems. Laplace domain analysis Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output

More information

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues.

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues. Similar Matrices and Diagonalization Page 1 Theorem If A and B are n n matrices, which are similar, then they have the same characteristic equation and hence the same eigenvalues. Proof Let A and B be

More information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

Solving Homogeneous Systems with Sub-matrices

Solving Homogeneous Systems with Sub-matrices Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State

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

Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

More information

The Jordan Canonical Form

The Jordan Canonical Form The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finite-dimensional vector space over an algebraically closed field. Here we develop

More information

Linear Algebra Highlights

Linear Algebra Highlights Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

More information