# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

5 Pages

### hw06

Course: MATH 25, Fall 2009
School: Harvard
Rating:

Word Count: 884

#### Document Preview

25B MATH PROBLEM SET #6 DUE TUESDAY MARCH 22ND Half of this assignment will be graded by Yan and the other half will be graded by Toly. Please turn in the problems from section 1 (which will be graded by Yan) separately from the problems from section 2 (which will be graded by Toly). Standing assumptions For this problem set, unless otherwise stated, you should work over the field C and assume that all vector...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Massachusetts >> Harvard >> MATH 25

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
25B MATH PROBLEM SET #6 DUE TUESDAY MARCH 22ND Half of this assignment will be graded by Yan and the other half will be graded by Toly. Please turn in the problems from section 1 (which will be graded by Yan) separately from the problems from section 2 (which will be graded by Toly). Standing assumptions For this problem set, unless otherwise stated, you should work over the field C and assume that all vector spaces are finite-dimensional. 1. Yan's problems (1) Computing things (a) Compute the Jordan canonical form of -3 2 0 -6 0 3 0 0 2 1 3 2 6 -3 0 9 (b) Let 2 1 1 A = 0 2 1 0 0 -1 Find a matrix B in Jordan canonical form and an invertible matrix Q such that B = QAQ-1 . (c) Are the matrices 0 2 0 2 1 0 2 0 0 0 2 0 and 0 0 2 0 0 2 similar? (2) Let T : V V be a linear transformation. Suppose that is an eigenvalue of T and that p is a polynomial. (a) Is p() an eigenvalue of p(T )? (b) Are all eigenvalues of p(T ) of this form? (3) Let A and B be n n matrices. Show that every non-zero eigenvalue of AB is an eigenvalue of BA, and conversely. (4) The set of invertible matrices is open (twice) (a) Show that the determinant of an n n matrix gives a continuous function from 2 2 Rn to R. Deduce that the set of invertible matrices is open in Rn . (b) State and prove a formula for the inverse of the matrix I -A which is valid when every entry of the matrix A is sufficiently small. Use this to 2 give another proof that the set of invertible matrices is open in Rn . Do the 1 1 matrix case first. In fact the set of invertible matrices is also dense in the set of all matrices. 2. Toly's problems (1) Matrix differential equations (a) Show that if A is an n n matrix, x1 , . . . xn are constant scalars, and y denotes the derivative of y with respect to t then x1 y1 . = exp(At) . . . . . yn xn is a solution to the matrix differential equation y1 y1 . = A . . . . . . yn yn (b) Suppose that y1 . . . () yn is any solution to (). Compute the derivative of y1 . exp(-At) . . yn with respect to t. Deduce that there exist constants x1 , . . . , xn such that x1 y1 . = exp(At) . . . . . . xn yn So the general solution of the system of linear differential equations () is as in (a). We can compute this efficiently by diagonalising A. (2) Higher-order linear differential equations and systems of linear equations (a) Suppose that the complex-valued function f (t) satisfies the differential equation f (n) + an-1 f (n-1) + . . . + a1 f (1) + a0 f = 0 where f (i) denotes the ith derivative of f . Write down a matrix-valued differential satisfied equation by the vector f f (1) . . . . f (n-1) (b) Use this to find all solutions to the differential equation d2 f = -f. dt2 (c) Show that the general real-valued solution to this differential equation is f (t) = A cos(t) + B sin(t) for some constants A, B R. This gives an algorithmic way to solve any linear n-th order differential equation with constant coefficients. (3) Jordan blocks and resonance Use the method above to find all real-valued solutions to the differential equations y - 2y - 3y = 0 and y - 2y + y = 0. (4) Autonomous dynamical systems In this problem we work in R2 , and so work over the field R. (a) Suppose that () x y =A x y where A is a 2 2 matrix with real eigenvalues, one strictly positive and one strictly negative. Show that solution trajectories t (x(t), y(t)) near the origin look like What are the two straight lines on this picture? This picture is called the "phase portrait" of the autonomous dynamical system (). It is called "autonomous" because the coefficients of () -- the entries of A -- do not vary with time. (b) Suppose that we change A. What are the other possible phase portraits? There are lots of possibilities. A could have both eigenvalues positive, or two complex eigenvalues, or . . . (5) A simple model of an epidemic A simple model of an epidemic in a city is as follows. Susceptible people enter the city at a constant rate of per day. Infected people recover or die after a certain number of days. If they recover, they are immune. The num...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Harvard - MATH - 25
MATH 25B PROBLEM SET #11 DUE TUESDAY 3RD MAY Half of this assignment will be graded by Yan and the other half will be graded by Toly. Please turn in the problems from section 1 (which will be graded by Toly) separately from the problems from section
Harvard - M - 213
Math 213a: Complex analysis Problem Set #8 (12 November 2003): Harmonic functions and their uses, contdFirst, an observation on the coecients of the linear equations used to determine the logarithm of our conformal map of a nitely connected region
Harvard - MATH - 25
MATH 25B PROBLEM SET #7 DUE FRIDAY MARCH 25TH1. Three problems (1) Square roots Does every matrix have a square root? In other words, if X is an n n matrix, must there equal A such that X = A2 ? And if A2 = B 2 , must A = B? (2) Orthogonal and un
Harvard - M - 21
Harvard - MATH - 25
MATH 25B PROBLEM SET #12 DUE WEDNESDAY 11TH MAY Half of this assignment will be graded by Yan and the other half will be graded by Toly. Please turn in the problems from section 1 (which will be graded by Yan) separately from the problems from secti
Harvard - M - 213
Math 213a: Complex analysis Problem Set #6 (29 October 2003): The Gamma function; univalent functions and normal families1. [Gauss multiplication formula] Let n be a positive integer, and definen-1F (z) =k=0z+k . ni) Show that F (z) has t
Harvard - MATH - 192
Collect homework; handout solutions and new problem sets Remind students: time spent, collaborators consulted Lectures: Tues. and Thurs., 2:30-4:00, Sever 103 Sections (optional): Mon., 5-6, Sci. Ctr. 309 My office hours: Tues. and Weds., 1:30-2:00,
Harvard - MATHE - 311
Solutions to Homework Problem Set 10 Problem (1) This problem concerned the 3 by 3 array of people who started off in random order in terms of height. Each column was ordered front to back shortest to tallest. Next each row was reordered from right t
Harvard - MATHE - 311
Answers to Second Problem SetMath E311 Spring 20081) First, critique the following proof by cases (i.e. is it a valid proof? are there holes in the logic? be sure to explain your answer carefully). Theorem: If x is any real number then x2 x. Pro
Harvard - MATH - 192
Collect homework; handout solutions and new problem sets It's a good idea to read the solutions, even if you got the problems right. A case in point is today's solution set, which contains tips on writing programs in Maple. From now on, email homewor
Harvard - MATH - 192
Take-home final exam due on Tuesday, Jan. 22 at 4 p.m. Questions (logistical or mathematical)? Recommended reading in GK&amp;P: pages 276-287 (Fibonacci numbers) pages 287-290 (continuants) TODAY: Frieze patterns and diamond patterns The weighted version
Harvard - MATH - 192
Arrive half-hour early Write on board: &quot;Prof. James Propp&quot; (call me Prof. Propp in the context of Math 192) Write on board: http:/www.math.harvard.edu/~propp/192.html www.fas.harvard.edu/courses/~math192 My goal is to inculcate two things: knowledge
Harvard - M - 21
PROBLEM 4(a) F For example, the system x=1, 2x=2 represented by Ax = b where A = [1 2] (a 2x1 matrix) [ 1 ] b = [ ] [ 2 ](b) F The map does not take zero to zero. (Reflection about
Harvard - M - 21
PROBLEM 1(a) We have |v1| = sqrt(1*1 + 3*3 + 3*3 + 9*9) = sqrt(100) = 10,so w1 = v1 / 10 = [1/10, 3/10, 3/10, 9/10]. Thenw1.v2 = (1*2 + 3*1 + 3*6 + 9*3) / 10 = 5, so the projectionof v2 to the orthogonal complement of the span of w1 isv2 - (
Harvard - M - 21
PROBLEM 3(a) Since the matrix is sparse we can easily find the determinant byexpanding by minors. In this case any row or column works equally well.Using the first row, we get [ c^2-2 0 0 ] [ 1 c^2-2 0
Harvard - M - 21
PROBLEM 2For the line y=ax+b to pass through the given points(-1,1), (0,2), (1,2), (2,0) the coefficients would haveto satisfy the (inconsistent) linear system -a + b = 1 b = 2 a + b = 2 2a + b = 0which in matrix form is [-1 1
Harvard - M - 21
PROBLEM 5(a) T We have B = S' A S where S is some invertible matrix and S' is the inverse of A. Therefore B is the product of invertible matrices, and so is itself invertible.(b) T Pythagoras: 3*3 + 4*4 = 5*5 so the dot produ
Harvard - M - 21
PROBLEM 2(a) To row-reduce the matrix [ 0 0 1 | 1 0 0 ] [ 0 2 1 | 0 1 0 ] [ 3 2 1 | 0 0 1 ]we switch rows I and III: [ 3 2 1 | 0 0 1 ] [ 0 2 1 | 0 1 0 ] [ 0 0 1 | 1 0 0 ]and multiply row I by (1/3): [ 1 2/3 1/3 | 0 0 1/3 ] [
Harvard - CS - 222
The Link Prediction Problem for Social NetworksDavid Liben-Nowell Laboratory for Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 USA dln@theory.lcs.mit.edu Jon Kleinberg Department of Computer Science Cornell University It
Harvard - CS - 225
CS 225: Pseudorandomness Problem Set 1Assigned: Tue. Feb. 6, 2007Prof. Salil VadhanDue: Wed. Feb. 21, 2007(1 PM) Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to
Harvard - CS - 225
CS225: PseudorandomnessProf. Salil VadhanLecture 4: Random WalksFebruary 13, 2007 Based on scribe notes by Dave Troiano and Brian Greenberg.1Graph ConnectivityOne of the most basic problems in computer science is that of deciding connectiv
Harvard - CS - 225
CS 225: Pseudorandomness Problem Set 3Assigned: Mar. 7, 2007Prof. Salil VadhanDue: Mar. 21, 2007 (1 PM) Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to Carol Har
Harvard - CS - 225
CS225: PseudorandomnessProf. Salil VadhanLecture 15: List-Decoding AlgorithmsApril 5, 2007 Based on scribe notes by xxxx. Let C be a code with encoding function Enc : {1, . . . , N } n . Given any received word r n , we would like to nd all
Harvard - CS - 225
CS225: PseudorandomnessProf. Salil VadhanLecture 14: Error-Correcting CodesApril 3, 2007 Based on scribe notes by Sasha Schwartz and Adi Akavia.1Basic DenitionsThe eld of coding theory is motivated by the problem of communicating reliably
Harvard - CS - 225
CS 225: Pseudorandomness Problem Set 5Assigned: Apr. 12, 2007Prof. Salil VadhanDue: Apr. 25, 2007 (1 PM) Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to Carol Ha
Harvard - CS - 225
CS 225: Pseudorandomness Problem Set 2Assigned: Feb. 20, 2007Prof. Salil VadhanDue: Mar. 7, 2007 (1 PM) Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to Carol Har
Harvard - CS - 225
CS 225: Pseudorandomness Problem Set 4Assigned: Mar. 22, 2007Prof. Salil VadhanDue: Apr. 11, 2007 (1 PM) Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to Carol Ha
Harvard - CS - 286
Evolution of Cooperative problem-solving in an artificial economyby E. Baum and I. Durdanovicpresented by Quang DuongOutline Reinforcement learning and other learning approaches' limitations Artificial Economy Representation Language: S-expre
Harvard - CS - 286
CS286r Multi-Agent Learning Homework 2: Mechanism Design and ImplementationSpring Term 2006 Prof. David Parkes Division of Engineering and Applied Sciences Harvard University Feb 6, 2006Due: Monday 2/27/2006, at the beginning of class. You may use
Harvard - CS - 286
CS286r Electronic Market Design Homework 2: Mechanism DesignSpring Term 2003 Prof. David Parkes Division of Engineering and Applied Sciences Harvard University Feb 13, 2003Due: Thursday 2/20/2003, in the beginning of class. You may use any sources
Harvard - CS - 286
On Partially Controlled Multi-Agent SystemsBy: Ronan I. Brafman and Moshe TennenholtzPresentation By: Katy Milkman CS286r - April 12, 2006CS 286r - April 12, 2006 1Partially Controlled Multi-Agent Systems (PCMAS) Controllable Agents: agents tha
Harvard - CS - 286
ParkesCS 286r1CS 286r: Electronic Market DesignDavid C. Parkesparkes@eecs.harvard.eduSpring, 2003ParkesCS 286r2Motivation Fast computers and high bandwidth has changed to cost of dynamic market mechanisms: automated winner-dete
Harvard - CS - 286
CS286r Computational Mechanism Design: Project Suggestions Prof. David Parkes, Spring 20071Class ProjectsThe goal of the nal project is to develop a deep understanding of an important research area and, to the extent possible, work on an open p
Harvard - CS - 700
Problem descriptionROI heuristicDynamics of the systemDiscussionDynamics of Bid Optimization in Online Advertisement AuctionsC. Borgs, J. Chayes, O. Etesami, N. Immorlica, K. Jain, M. Mahdian By Ludk Cigler &amp; Thomas Laut e e eOctober 21,
Harvard - CS - 286
ParkesInteger Programming1'\$Integer ProgrammingDavid C. ParkesDivision of Engineering and Applied Science, Harvard UniversityCS 286rSpring 2002&amp;%ParkesInteger Programming2'\$Motivation Very flexible and expressive model
Harvard - CS - 286
ParkesMechanism Design1'\$Linear ProgrammingDavid C. ParkesDivision of Engineering and Applied Science, Harvard UniversityCS 286rSpring 2002&amp;%ParkesMechanism Design2'\$Introduction LP is the problem of optimizing a linea
Harvard - CS - 286
ParkesMechanism Design1'\$Mechanism Design IIDavid C. ParkesSchool of Engineering and Applied Science, Harvard UniversityCS 286rSpring 2007&amp;%ParkesMechanism Design2'\$Positive &amp; Negative Results We have seen two positive
Harvard - CS - 286
ParkesMechanism Design1'\$Classic Mechanism Design (III)David C. ParkesDivision of Engineering and Applied Science, Harvard UniversityCS 286rSpring 2002&amp;%ParkesMechanism Design2'Vickrey-Clarke-Groves Mechanism(VCG or &quot;Piv
Harvard - CS - 286
Harvard - CS - 141
CS141SyllabusComputer Science 141: Computing Hardware Course Information Fall 2008September 14, 20081OutlineThe main emphasis of this course is on the basic concepts of digital computing hardware and fundamental digital design principles
Harvard - CS - 286
Betting on PermutationsBetting on PermutationsBrett HarrisonHarvard UniversityOctober 19, 2008Betting on Permutations IntroductionIntroductionLast week, we looked at optimal strategies for traderse.g. when to buy a security, what type of
Harvard - IWDDS - 06
Network-Aware Overlays with Network CoordinatesPeter Pietzuch, Jonathan Ledlie, Michael Mitzenmacher, Margo Seltzer Harvard University, Cambridge, MA, USA hourglass@eecs.harvard.eduAbstractNetwork coordinates, which embed network distance measure
Harvard - CS - 250
Notes on Cyclone Extended Static CheckingGreg MorrisettHarvard UniversityStatic Extended Checking: SEX-CSimilar approach to ESC-M3/Java: Calculate a 1st-order predicate describing the machine state at each program point. Generate verification
Harvard - CS - 243
Homework April 18, due April 25 1. Read chapters 20, 21
Harvard - CS - 243
Homework, April 4, due April 111. read chapters 13 and 142. (2 pts) Book problem 13-13. (2 pts) Book problem 13-24. (2 pts) Book problem 13-35. (2 pts) Book problem 13-46. (3 pts) Book problem 13-57. (2 pts) Compare the security properties of
Harvard - CS - 266
CS266 final paper guidelines:=Logistics: -The final paper is due monday Jan 10 by noon. Email the paper to me atcs266-reviews@eecs.harvard.edu. The paper must be in * PDF * format.Papers Statistics: --The papers must be no more than 10 page
Harvard - SB - 301
Prolog toEngineering in the Biological Substrate: Information Processing in Genetic CircuitsAn introduction to the paper by Simpson, Cox, Peterson, and SaylerIf one thinks of gene circuits and networks as methods for processing biological inform
Harvard - SB - 301
Engineering in the Biological Substrate: Information Processing in Genetic CircuitsMICHAEL L. SIMPSON, SENIOR MEMBER, IEEE, CHRIS D. COX, GREGORY D. PETERSON, SENIOR MEMBER, IEEE, AND GARY S. SAYLER Contributed PaperWe review the rapidly evolving e
Harvard - SB - 301
Harvard - SB - 301
Development 127, 2977-2987 (2000) Printed in Great Britain The Company of Biologists Limited 2000 DEV25662977REVIEW ARTICLE Measuring dimensions: the regulation of size and shapeStephen J. Day1 and Peter A. Lawrence2128 St Oswalds Road, York,
Harvard - SB - 301
Harvard - SB - 301
ARTICLE IN PRESSJournal of Theoretical Biology 235 (2005) 431449 www.elsevier.com/locate/yjtbiRobustness and fragility of Boolean models for genetic regulatory networks Madalena Chavesa, Reka Albertb, Eduardo D. SontagabDepartment of Mathemat
Harvard - SB - 301
Cell, Vol. 100, 7988, January 7, 2000, Copyright 2000 by Cell PressMolecular VitalismReviewMarc Kirschner,* John Gerhart, and Tim Mitchison* * Department of Cell Biology Harvard Medical School Boston, Massachusetts 02114 Department of Molecula
Harvard - SB - 301
Proc. Natl. Acad. Sci. USA Vol. 95, pp. 84208427, July 1998Perspective EvolvabilityMarc Kirschner* and John Gerhart*Department of Cell Biology, Harvard Medical School, Boston, MA 02115; and Department of Molecular and Cell Biology, University of
Harvard - CS - 266
CS266 Final Paper ACO Routing in Wireless Sensor Networks Atanu Roy Chowdhury Jason Waterman January 14th, 2008AbstractThe recent popularity of applications based on wireless sensor networks provides a strong motivation for pushing its technologic
Harvard - CS - 266
Lessons from Regenerative Systems in Biology Harvard/FAS/CS266 Project PaperAlex Shpunt and Seth Frey 13-January-2008Project ObjectiveThe objective of the project was to investigate models for canonical regenerative systems in biology (Hydra, Sta
Harvard - SB - 301
letters to nature.Dynamic control of positional information in the early Drosophila embryoJohannes Jaeger1, Svetlana Surkova2, Maxim Blagov2, Hilde Janssens1, David Kosman3, Konstantin N. Kozlov2, Manu1, Ekaterina Myasnikova2, Carlos E. Vanario-A
Harvard - CS - 266
Ant-Based ComputingLoizos MichaelDivision of Engineering and Applied Sciences Harvard University, Cambridge, MA 02138, U.S.A. loizos@eecs.harvard.eduAbstract. We propose a biologically and physically plausible model for ants and pheromones, and s
Harvard - SB - 301
504OpinionTRENDS in Genetics Vol.18 No.10 October 2002Conservation of the segmented germband stage: robustness or pleiotropy?Frietson Galis, Tom J.M. van Dooren and Johan A.J. MetzGene expression patterns of the segment polarity genes in the
Harvard - CS - 266
Dynamic Shape FormationZain Khalid and Vaidya Rajagopalan CS 266 Abstract: In this paper we are investigating dynamic shape formation, with the aim of realistically simulating dynamic maneuvers commonly exhibited in nature. These simulations were bu
Harvard - SB - 301
letters to natureMeasurementsThe number of progeny was highly variable between families and environments. Twentynine families consisting of 12 sires mated to 2 or 3 dams provided sufcient progeny (ve male and ve female) from all food treatments. St