Example: A Markov Process
|
|
- Jack Basil Mills
- 6 years ago
- Views:
Transcription
1 Example: A Markov Process Divide the greater metro region into three parts: city such as St. Louis), suburbs to include such areas as Clayton, University City, Richmond Heights, Maplewood, Kirkwood,...) and exurbs the far out areas where people associated with the metro area might live: for example St. Charles county, Jefferson County,...) In this oversimplified model, we ignore people entering/leaving the region; we ignore births/deaths. We assume that the total population is constant and that people move around between the city, suburbs and exurbs. Data collection lets us estimate how the population shifts from year to year: Suppose the transition matrix E is: Eœ Moving From C S E Æ Æ Æ Þ& Þ"! Þ"! C City) Þ"! Þ'! Þ! to Ä S Suburbs) Þ"& Þ! Þ! E Exurbs) Going down Column 1 tells us what proportion %) of the C population will move to C, S, E during the year. Column 1 accounts for where the whole city population lives at the end of a year, so Column 1 adds up to " 100%). Similarly, SumColumn 2) œ " and SumColumn 3) œ"þ Suppose the metro region has a total population of 2,000,000 distributed as: %!!!!! "%!!!!!!!!!! City Suburb Exurbs After one year the new distribution is Þ& Þ"! Þ"! %!!!!! %'!!!! Þ"! Þ'! Þ! "%!!!!! œ *!!!! Þ"& Þ! Þ!!!!!! '!!!! City Suburb Exurbs To avoid working with such large numbers, we will use a population state vector written instead with proportions: for the initial state vector B! ß the people in the region the are distributed in proportions as %!!!!!!!!!!! "%!!!!!!!!!!!!!!!!!!!!!! Þ! œ Þ! œ Þ"! B! % C % S % E Then B Þ& Þ"! Þ"! Þ! Þ œeb œ Þ"! Þ'! Þ! Þ! œ Þ%' Þ"& Þ! Þ! Þ"! Þ" "! % C % S % E is next
2 year's population distribution. After 2 years, the population state distribution is Þ& Þ"! Þ"! Þ Þ%*& B œeb" œeðeb! ÑÑœEB! œ Þ"! Þ'! Þ! Þ%' œ Þ'"! Þ"& Þ! Þ! Þ" Þ)*& and so on: B8" œeb8 œeðeb8" Ñ œeðeðeb8 ÑÑœÞÞÞœE 8" B! **) Each state vector B8, after the initial vector B!ß is obtained from the preceding one by multiplication by E. Notice that: 1) We can also think of the proportions in E and in B! as probabilities: for example, the probability is!þ! that a randomly chosen person from the suburbs will move to the exurbs by next year. The!Þ! in the second row of B! tells us that the probability is 0.70 that a randomly chosen person out of the in the region) lives in the suburbs. 2) The entries in E and B! are nonnegative numbers, and the columns add up to ". A square matrix E satisfying these two conditions is called a stochastic matrix, and such a vector B! is called a probability vector. As we compute, each succeeding state vector B" ß B ß ÞÞÞß B8ß ÞÞÞ is still a probability vector we check this for the case, but the argument in the case of an 8 8 stochastic matrix is completely similar): +, - B If Eœ. / 0 is a stochastic matrix and C is a probability vector, then D +, - B +B,C-D. / 0 C œ.b/c0d. The entries are still 0, and they still sum to " À D 1B2C3D Ð+.1ÑBÐ,/2ÑCÐ-03ÑD œ" B" C" D œbcd œ" A system consisting of a stochastic matrix, an initial state probability vector B8" œ E B8 is called a Markov process. B! and an equation In a Markov process, each successive state B8" depends only on the preceding state B8Þ An important question about a Markov process is What happens in the long-run?, that is, what happens to as 8Ä? B 8 In our example, we can start with a good guess. Using Matlab, I quickly) computed
3 Þ)%' Þ)& "! "!! B"! œ E B! œ Þ)', ÞÞÞ ß B"!! œ E B! œ Þ)& and that Þ%* Þ%)' Þ)& "!!! B"!!! œe B! œ Þ)& The results are rounded to 4 decimal places; during Þ%)' displayed the calculations, Matlab carried along many more decimal places although even then small roundoffs were made.) Assuming the transition matrix and other modeling assumptions remain valid as time passes, it Þ)& seems like the population distribution moves toward a steady state B œ Þ)& with Þ%)' )Þ&% of the population in each of the city and suburbs, and with %Þ)'% in the exurbs. Using our knowledge of linear algebra, we can actually find this steady state B without repeated computations and guessing. Is there a steady state probability vector B for which EB œ B? equivalently, for which ÐE MÑB œ!? We begin by finding all Bthat satisfy the equation. Then among those solutions, we find an B that is also a probability vector. Þ& Þ"! Þ"! Here, EM œ Þ"! Þ%! Þ! Þ To avoid any roundoff error, we can convert to Þ"& Þ! Þ! fractions and row reduce the augmented matrix for ÐE MÑB œ! À " " "!! % "! "! % "! "! "!! " %! œ " "! µ ÞÞÞ µ! "! "! "! "! "! & & "&!!!!!! "!! "! "! " " "! "! "! The general solution is B œb where B is free. A more convenient rescaled) form is " B œ=. From among the solutions Bßwe want the one that is a probability vector. We also " " get it by choosing =œ, so that œ B œ has entries that add to "Þ This is the one and only probability vector that is a solution to EB œ B so is the steady state vector for the Markov process in our example.
4 Rounded to 4 decimal places : Þ)& œ Þ)&, the result estimated using Matlab. Þ%)' This is exactly what happens in many cases with a Markov process. The following is a result proven in courses that treat Markov processes in detail. Definition An 8 8stochastic matrix Eis called regular if for some positive integer, 5 the entries in the power E are all! not merely 0 ). " In the example above, Eis regular because EœE has all entries!þ Theorem If E is a regular 8 8stochastic matrix, then i) there exists a unique steady state probability vector B, that is, a probability vector for which EB œ B and ii) B8" œeb8 ÄB as 8Ä and this is true no matter which probability vector is used as the initial state B!.
5 The little that we already know about diagonalization and eigenvectors also sheds some light on this Markov process because the matrix Ehappens to be diagonalizable. Recall that: Definition A nonzero is called an eigenvector of the 8 8matrix E if E@ œ -@ for some scalar -. The scalar - is called an eigenvalue of EÐassociated with the is an eigenvector of any 8 8matrix) Ewith eigenvalue -. Then E@ œ œ EÐE@ Ñ œ EÐ-@ Ñ œ -EÐ@ Ñ œ -Ð-@ Ñ œ œ Ñ œ Ñ œ - EÐ@ Ñ œ - -@ œ ã 8 8 œ ÞÞÞ œ Þ& Þ"! Þ"! For our example, Eœ Þ"! Þ'! Þ! with Þ"& Þ! Þ! " corresponding eigenvalues - " œ! œ!þ'&, - œ "ß - œ & œ!þ%! and these eigenvectors form a basis for so E is diagonalizable. In the recent supplementary homework, we saw a method for finding eigenvalues -: find the -'s if any) that make det ÐE -MÑ œ!þ Actually applying the method for a matrix E leads to a cubic equation that must be solved for - this can be done, but may be difficult depending on the matrix E in general. If we find any eigenvalues -, then we can solve EB œ -Bto find the corresponding eigenvectors. For the example, I used Matlab to help find the diagonal factorization for E: Þ& Þ"! Þ"! Þ'&!! Eœ Þ"! Þ'! Þ! œt! "! T ", where Þ"& Þ! Þ!!! Þ%! Þ)""" Þ%)&"! Tœ Þ%% Þ%)&" Þ"" Þ%)' Þ' Þ!" rounded to 4 decimal places) The columns of T are approximately) the ß@ that form a basis for, and the eigenvalues ß are the entries in the diagonal matrix: Þ'&ß "ß and Þ% Since the eigenvectors are a basis for, the initial state vector B! can be written as a linear combination of the basis eigenvectors: Þ! B! œ Þ! œ -"@" Ðwe could find the weights -ß-ß- " if neededñ Þ"!
6 Therefore B œeb Ñœ-" E@ - E@ - E@ œ "! " " " " "! " " " B œ E B œ EÐ- - - Ñ œ - - E@ - - E@ - - E@ " " " œ -"- "@" - ã B œ E B œ ÞÞÞ ÞÞÞ œ ! " " " Therefore we can see that " " œ Ä 8Ä " " lim B œ lim Ð- - - Ñ œ lim - Ñ œ 8Ä " If we had actually found the weights -ß-ß- " above then we would now know the steady state probability vector B Ðbecause we would know - Ñ Þ But, instead, we can still find B, because we know Bis supposed to be a probability Þ%)&" is approximately) the second column of Tœ Þ%)&" and Þ' for B to be a probability vector, the sum of the entries must be "À " this requires use to choose - œ sum of the entries œ " Þ%)&" Þ%)&" Þ' Þ&)*! Þ%)&" Þ)& So B Þ&)*! Þ%)&" œ Þ)& Ðrounded to 4 decimal places) Þ' Þ%)' œ the same steady state B we found earlier. So: diagonalization can help in understanding Markov Processes and similar kinds of linear difference equations.
The Spotted Owls 5 " 5. = 5 " œ!þ")45 Ð'!% leave nest, 30% of those succeed) + 5 " œ!þ("= 5!Þ*%+ 5 ( adults live about 20 yrs)
The Spotted Owls Be sure to read the example at the introduction to Chapter in the textbook. It gives more general background information for this example. The example leads to a dynamical system which
More informationHere are proofs for some of the results about diagonalization that were presented without proof in class.
Suppose E is an 8 8 matrix. In what follows, terms like eigenvectors, eigenvalues, and eigenspaces all refer to the matrix E. Here are proofs for some of the results about diagonalization that were presented
More informationIntroduction to Diagonalization
Introduction to Diagonalization For a square matrix E, a process called diagonalization can sometimes give us more insight into how the transformation B ÈE B works. The insight has a strong geometric flavor,
More information: œ Ö: =? À =ß> real numbers. œ the previous plane with each point translated by : Ðfor example,! is translated to :)
â SpanÖ?ß@ œ Ö =? > @ À =ß> real numbers : SpanÖ?ß@ œ Ö: =? > @ À =ß> real numbers œ the previous plane with each point translated by : Ðfor example, is translated to :) á In general: Adding a vector :
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationCity 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 informationThe Leontief Open Economy Production Model
The Leontief Open Economy Production Model We will look at the idea for Leontief's model of an open economy a term which we will explain below. It starts by looking very similar to an example of a closed
More informationThe Leontief Open Economy Production Model
The Leontief Open Economy Production Model We will look at the idea for Leontief's model of an open economy a term which we will explain below. It starts by looking very similar to an example of a closed
More informationsolve EB œ,, we can row reduce the augmented matrix
Motivation To PY Decomposition: Factoring E œ P Y solve EB œ,, we can row reduce the augmented matrix Ò + + ÞÞÞ+ ÞÞÞ + l, Ó µ ÞÞÞ µ Ò- - ÞÞÞ- ÞÞÞ-. Ó " # 3 8 " # 3 8 we get to Ò-" -# ÞÞÞ -3 ÞÞÞ-8Ó in an
More informationSystems of Equations 1. Systems of Linear Equations
Lecture 1 Systems of Equations 1. Systems of Linear Equations [We will see examples of how linear equations arise here, and how they are solved:] Example 1: In a lab experiment, a researcher wants to provide
More informationÆ Å not every column in E is a pivot column (so EB œ! has at least one free variable) Æ Å E has linearly dependent columns
ßÞÞÞß @ is linearly independent if B" @" B# @# ÞÞÞ B: @: œ! has only the trivial solution EB œ! has only the trivial solution Ðwhere E œ Ò@ @ ÞÞÞ@ ÓÑ every column in E is a pivot column E has linearly
More informationsolve EB œ,, we can row reduce the augmented matrix
PY Decomposition: Factoring E œ P Y Motivation To solve EB œ,, we can row reduce the augmented matrix Ò + + ÞÞÞ+ ÞÞÞ + l, Ó µ ÞÞÞ µ Ò- - ÞÞÞ- ÞÞÞ-. Ó " # 3 8 " # 3 8 When we get to Ò-" -# ÞÞÞ -3 ÞÞÞ-8Ó
More informationB œ c " " ã B œ c 8 8. such that substituting these values for the B 3 's will make all the equations true
System of Linear Equations variables Ð unknowns Ñ B" ß B# ß ÞÞÞ ß B8 Æ Æ Æ + B + B ÞÞÞ + B œ, "" " "# # "8 8 " + B + B ÞÞÞ + B œ, #" " ## # #8 8 # ã + B + B ÞÞÞ + B œ, 3" " 3# # 38 8 3 ã + 7" B" + 7# B#
More informationand 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 informationDefinition Suppose M is a collection (set) of sets. M is called inductive if
Definition Suppose M is a collection (set) of sets. M is called inductive if a) g M, and b) if B Mß then B MÞ Then we ask: are there any inductive sets? Informally, it certainly looks like there are. For
More informationEigenvalue and Eigenvector Homework
Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues
More informationPART I. Multiple choice. 1. Find the slope of the line shown here. 2. Find the slope of the line with equation $ÐB CÑœ(B &.
Math 1301 - College Algebra Final Exam Review Sheet Version X This review, while fairly comprehensive, should not be the only material used to study for the final exam. It should not be considered a preview
More information8œ! This theorem is justified by repeating the process developed for a Taylor polynomial an infinite number of times.
Taylor and Maclaurin Series We can use the same process we used to find a Taylor or Maclaurin polynomial to find a power series for a particular function as long as the function has infinitely many derivatives.
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
More informationSIMULATION - PROBLEM SET 1
SIMULATION - PROBLEM SET 1 " if! Ÿ B Ÿ 1. The random variable X has probability density function 0ÐBÑ œ " $ if Ÿ B Ÿ.! otherwise Using the inverse transform method of simulation, find the random observation
More informationLecture 15, 16: Diagonalization
Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose
More informationLinear 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 informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More information1. (3pts) State three of the properties of matrix multiplication.
Math 125 Exam 2 Version 1 October 23, 2006 60 points possible 1. (a) (3pts) State three of the properties of matrix multiplication. Solution: From page 72 of the notes: Theorem: The Properties of Matrix
More informationACT455H1S - TEST 2 - MARCH 25, 2008
ACT455H1S - TEST 2 - MARCH 25, 2008 Write name and student number on each page. Write your solution for each question in the space provided. For the multiple decrement questions, it is always assumed that
More informationMATH 310, REVIEW SHEET 2
MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
More informationThis operation is - associative A + (B + C) = (A + B) + C; - commutative A + B = B + A; - has a neutral element O + A = A, here O is the null matrix
1 Matrix Algebra Reading [SB] 81-85, pp 153-180 11 Matrix Operations 1 Addition a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn + b 11 b 12 b 1n b 21 b 22 b 2n b m1 b m2 b mn a 11 + b 11 a 12 + b 12 a 1n
More informationSTAT 2122 Homework # 3 Solutions Fall 2018 Instr. Sonin
STAT 2122 Homeork Solutions Fall 2018 Instr. Sonin Due Friday, October 5 NAME (25 + 5 points) Sho all ork on problems! (4) 1. In the... Lotto, you may pick six different numbers from the set Ö"ß ß $ß ÞÞÞß
More informationLinear 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 informationRecitation 8: Graphs and Adjacency Matrices
Math 1b TA: Padraic Bartlett Recitation 8: Graphs and Adjacency Matrices Week 8 Caltech 2011 1 Random Question Suppose you take a large triangle XY Z, and divide it up with straight line segments into
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationA&S 320: Mathematical Modeling in Biology
A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~dmu228/as320/ Spring 2016 David Murrugarra (University of Kentucky) A&S
More informationMIT Final Exam Solutions, Spring 2017
MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of
More information= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E.
3.3 Diagonalization Let A = 4. Then and are eigenvectors of A, with corresponding eigenvalues 2 and 6 respectively (check). This means 4 = 2, 4 = 6. 2 2 2 2 Thus 4 = 2 2 6 2 = 2 6 4 2 We have 4 = 2 0 0
More informationHomework 3 Solutions Math 309, Fall 2015
Homework 3 Solutions Math 39, Fall 25 782 One easily checks that the only eigenvalue of the coefficient matrix is λ To find the associated eigenvector, we have 4 2 v v 8 4 (up to scalar multiplication)
More informationMatrices related to linear transformations
Math 4326 Fall 207 Matrices related to linear transformations We have encountered several ways in which matrices relate to linear transformations. In this note, I summarize the important facts and formulas
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationThe Hahn-Banach and Radon-Nikodym Theorems
The Hahn-Banach and Radon-Nikodym Theorems 1. Signed measures Definition 1. Given a set Hand a 5-field of sets Y, we define a set function. À Y Ä to be a signed measure if it has all the properties of
More information22m: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 informationInner Product Spaces
Inner Product Spaces In 8 X, we defined an inner product? @? @?@ ÞÞÞ? 8@ 8. Another notation sometimes used is? @? ß@. The inner product in 8 has several important properties ( see Theorem, p. 33) that
More informationMAT1302F Mathematical Methods II Lecture 19
MAT302F Mathematical Methods II Lecture 9 Aaron Christie 2 April 205 Eigenvectors, Eigenvalues, and Diagonalization Now that the basic theory of eigenvalues and eigenvectors is in place most importantly
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationChapter 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 informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationSuggestions - Problem Set (a) Show the discriminant condition (1) takes the form. ln ln, # # R R
Suggetion - Problem Set 3 4.2 (a) Show the dicriminant condition (1) take the form x D Ð.. Ñ. D.. D. ln ln, a deired. We then replace the quantitie. 3ß D3 by their etimate to get the proper form for thi
More information4.1 Markov Processes and Markov Chains
Chapter Markov Processes. Markov Processes and Markov Chains Recall the following example from Section.. Two competing Broadband companies, A and B, each currently have 0% of the market share. Suppose
More informationRemark 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 informationProofs Involving Quantifiers. Proof Let B be an arbitrary member Proof Somehow show that there is a value
Proofs Involving Quantifiers For a given universe Y : Theorem ÐaBÑ T ÐBÑ Theorem ÐbBÑ T ÐBÑ Proof Let B be an arbitrary member Proof Somehow show that there is a value of Y. Call it B œ +, say Þ ÐYou can
More informationElementary Linear Algebra Review for Exam 3 Exam is Friday, December 11th from 1:15-3:15
Elementary Linear Algebra Review for Exam 3 Exam is Friday, December th from :5-3:5 The exam will cover sections: 6., 6.2, 7. 7.4, and the class notes on dynamical systems. You absolutely must be able
More informationMath 2J Lecture 16-11/02/12
Math 2J Lecture 16-11/02/12 William Holmes Markov Chain Recap The population of a town is 100000. Each person is either independent, democrat, or republican. In any given year, each person can choose to
More informationTOTAL DIFFERENTIALS. In Section B-4, we observed that.c is a pretty good approximation for? C, which is the increment of
TOTAL DIFFERENTIALS The differential was introduced in Section -4. Recall that the differential of a function œ0ðñis w. œ 0 ÐÑ.Þ Here. is the differential with respect to the independent variable and is
More informationEigenvalues and Eigenvectors
November 3, 2016 1 Definition () The (complex) number λ is called an eigenvalue of the n n matrix A provided there exists a nonzero (complex) vector v such that Av = λv, in which case the vector v is called
More information5 Eigenvalues and. Eigenvectors. Dynamical Systems and Spotted Owls
5 Eigenvalues and Eigenvectors INTROUCTORY EXAMPLE ynamical Systems and Spotted Owls In 99, the northern spotted owl became the center of a nationwide controversy over the use and misuse of the majestic
More informationExample. We can represent the information on July sales more simply as
CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS 11 Matrices and Vectors In many occasions, we can arrange a number of values of interest into an rectangular array For example: Example We can
More informationRelations. Relations occur all the time in mathematics. For example, there are many relations between # and % À
Relations Relations occur all the time in mathematics. For example, there are many relations between and % À Ÿ% % Á% l% Ÿ,, Á ß and l are examples of relations which might or might not hold between two
More information1 Principal component analysis and dimensional reduction
Linear Algebra Working Group :: Day 3 Note: All vector spaces will be finite-dimensional vector spaces over the field R. 1 Principal component analysis and dimensional reduction Definition 1.1. Given an
More informationComputationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:
Diagonalization We have seen that diagonal and triangular matrices are much easier to work with than are most matrices For example, determinants and eigenvalues are easy to compute, and multiplication
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationDIAGONALIZATION. 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[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
More informationExamples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.
The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section
More informationLinear Algebra Basics
Linear Algebra Basics For the next chapter, understanding matrices and how to do computations with them will be crucial. So, a good first place to start is perhaps What is a matrix? A matrix A is an array
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationExample Let VœÖÐBßCÑ À b-, CœB - ( see the example above). Explain why
Definition If V is a relation from E to F, then a) the domain of V œ dom ÐVÑ œ Ö+ E À b, F such that Ð+ß,Ñ V b) the range of V œ ran( VÑ œ Ö, F À b+ E such that Ð+ß,Ñ V " c) the inverse relation of V œ
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Recall: Parity of a Permutation S n the set of permutations of 1,2,, n. A permutation σ S n is even if it can be written as a composition of an
More informationEigenvalues 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 informationThe 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 informationEigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems
More information22m:033 Notes: 3.1 Introduction to Determinants
22m:033 Notes: 3. Introduction to Determinants Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman October 27, 2009 When does a 2 2 matrix have an inverse? ( ) a a If A =
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationQuestion: 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 informationDiagonalization. Hung-yi Lee
Diagonalization Hung-yi Lee Review If Av = λv (v is a vector, λ is a scalar) v is an eigenvector of A excluding zero vector λ is an eigenvalue of A that corresponds to v Eigenvectors corresponding to λ
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More information2. Let 0 be as in problem 1. Find the Fourier coefficient,&. (In other words, find the constant,& the Fourier series for 0.)
Engineering Mathematics (ESE 317) Exam 4 April 23, 200 This exam contains seven multiple-choice problems worth two points each, 11 true-false problems worth one point each, and some free-response problems
More informationBanded symmetric Toeplitz matrices: where linear algebra borrows from difference equations. William F. Trench Professor Emeritus Trinity University
Banded symmetric Toeplitz matrices: where linear algebra borrows from difference equations William F. Trench Professor Emeritus Trinity University This is a lecture presented the the Trinity University
More informationLecture 12: Diagonalization
Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Eigenvalues and Eigenvectors A an n n matrix of real numbers. The eigenvalues of A are the numbers λ such that Ax = λx for some nonzero vector x
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationMATH 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 informationCAAM 335 Matrix Analysis
CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points
More informationMATH 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 informationTherefore, 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 informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationFrom Handout #1, the randomization model for a design with a simple block structure can be written as
Hout 4 Strata Null ANOVA From Hout 1 the romization model for a design with a simple block structure can be written as C œ.1 \ X α % (4.1) w where α œðα á α > Ñ E ÐÑœ % 0 Z œcov ÐÑ % with cov( % 3 % 4
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationMath Camp Notes: Linear Algebra II
Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other
More informationChapter 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 informationMAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
More informationQUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, MAY 9, Examiners: Drs. K. M. Ramachandran and G. S. Ladde
QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, MAY 9, 2015 Examiners: Drs. K. M. Ramachandran and G. S. Ladde INSTRUCTIONS: a. The quality is more important than the quantity. b. Attempt
More informationJordan Canonical Form Homework Solutions
Jordan Canonical Form Homework Solutions For each of the following, put the matrix in Jordan canonical form and find the matrix S such that S AS = J. [ ]. A = A λi = λ λ = ( λ) = λ λ = λ =, Since we have
More informationLecture 10: Powers of Matrices, Difference Equations
Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each
More informationSection 1.3 Functions and Their Graphs 19
23. 0 1 2 24. 0 1 2 y 0 1 0 y 1 0 0 Section 1.3 Functions and Their Graphs 19 3, Ÿ 1, 0 25. y œ 26. y œ œ 2, 1 œ, 0 Ÿ " 27. (a) Line through a!ß! band a"ß " b: y œ Line through a"ß " band aß! b: y œ 2,
More informationMath 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 information6 EIGENVALUES AND EIGENVECTORS
6 EIGENVALUES AND EIGENVECTORS INTRODUCTION TO EIGENVALUES 61 Linear equations Ax = b come from steady state problems Eigenvalues have their greatest importance in dynamic problems The solution of du/dt
More informationFinal. for Math 308, Winter This exam contains 7 questions for a total of 100 points in 15 pages.
Final for Math 308, Winter 208 NAME (last - first): Do not open this exam until you are told to begin. You will have 0 minutes for the exam. This exam contains 7 questions for a total of 00 points in 5
More informationUnit 2, Section 3: Linear Combinations, Spanning, and Linear Independence Linear Combinations, Spanning, and Linear Independence
Linear Combinations Spanning and Linear Independence We have seen that there are two operations defined on a given vector space V :. vector addition of two vectors and. scalar multiplication of a vector
More information