PPT-Page Rank, Graph Eigenvalues, Kleinberg

Presentations  Nikhil Baradwaj and Seung Hwan Lee  1132017 Starting with  Page Rank  and the founders of Google it has become popular to model the web and social media as . 1 Introduction to Eigenvalues Linear equations come from steady state problems Eigenvalues have their greatest importance in dynamic problems The solution of dt is changing with time growing or decaying or oscillating We cant 64257nd it by eliminat Section 4.4. Eigenvalues and the Characteristic Polynomial. Characteristic Polynomial. If . A. is an . matrix the . characteristic polynomial . is a function of the variable . t. we call . that is the determinant of . Review of properties of vibration and buckling modes. What is nice about them?. Sensitivities of eigenvalues are really cheap! . Sensitivities of . eigevectors. . . Why bother getting them? . Think of where you want your car to have the least vibrations. deconvolution. as a general method to distinguish direct dependencies in networks . MIT group; Accepted Jun. 2013; Nature Biotechnology. Presented by Haicang Zhang. Feb.24 2013. Outline. Motivation. . can. be . interpreted. as a file of data. A . matrix. . is. a . collection. of . vectors. and . can. be . interpreted. as a data . base. The. red . matrix. . contain. . three. . column. 4. The. . Gauß. . scheme. A . linear. system of . equations. Matrix. algebra . deals. . essentially. . with. . linear. . linear. systems.. Multiplicative. . elements. .. A . non-linear. system. Hung-yi Lee. Chapter 5. In chapter 4, we already know how to consider a function from different aspects (coordinate system). Learn how to find a “good” coordinate system for a function. Scope. : Chapter 5.1 – 5.4. EIGEN … THINGS. (values, vectors, spaces … ). CONVENTION: . From now on, unless otherwise spec-. ified. , all matrices shall be square, i.e. . . . Another, less simple example:. . What are these . Prepared by Vince Zaccone. For Campus Learning Assistance Services at UCSB. Prepared by Vince Zaccone. For Campus Learning Assistance Services at UCSB. Consider the equation . , where A is an . nxn. MAT 275. A . linear system . is two or more linear equations in two or more variables taken together.. For example, . is a system of two linear equations in two variables.. A . solution of a system . Thomas Vidick. California Institute of Technology. . Joint work with . Itai Arad (Technion),. Zeph Landau and Umesh Vazirani (UC Berkeley). Local Hamiltonians.  .  .  .  .  .  .  .  .  . eigenvalues. Case II: Complex Eigenvalues. MAT 275. Recall that . . We will use this identity when solving systems of differential equations with constant coefficients in which the eigenvalues are complex. . Example: . A AUn fx CC C C C a U ucu o o r ox0000 r k ax CC C 2k n 2k a a al a a a U U UCU W UUCCU WW U CC a aCC C a CC ARZQORDGHGIURPKWWSVZZZFDPEULGJHRUJFRUH2FWDWVXEMHFWWRWKHDPEULGJHRU Thm.[B] LetX1;X2;


;Xkbeeigenvectorscorrespondingto distinct eigenvalues1;2;


;kofA.ThenfX1;X2;


;Xkgis linearlyindependent . Proof.AssumethatfX1;X2;
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