PDF-Linear Algebra part Eigenvalues and Eigenvectors by Evan Dummit v

100 Contents 1EigenvaluesandEigenvectors 11 The Basic Setup 1 12 Some Slightly More Advanced Results Ab out Eigenvalues 4 13 Theory of Similarity. 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 100 Contents 1 Systems of FirstOrder Linear Dierential Equations 1 11 General Theory of FirstOrder Linear Systems 1 12 Eigenvalue Metho d for diagonalizable co ecient matrices 3 13 Eigenvalue Metho Calculus Functions of single variable Limit con tinuity and differentiability Mean value theorems Evaluation of definite and improper integrals Partial derivatives Total derivative Maxima and minima Gradient Divergence and Curl Vector identities Di Let be some operator and a vector If does not change the direction of the vector is an eigenvector of the operator satisfying the equation 1 where is a real or complex number the eigenvalue corresponding to the eigenvector Thus the operator will o Diagonalization brPage 2br Eigenvalues and eigenvectors of an operator De64257nition Let be a vector space and be a linear operator A number is called an eigenvalue of the operator if for a nonzero vector The vector is called an eigenvector of as Calculus Functions of single variable Limit continuity and differentiability Mean value theorems Evaluation of definite and improper integrals Partial derivatives Total derivative Maxima and minima Gradient Divergence and Cu rl Vector identities Di BY. YAN RU LIN. SCOTT HENDERSON. NIRUPAMA GOPALASWAMI. GROUP 4. 11.1 EIGENVALUES & EIGENVECTORS. Definition. An . eigenvector. of a . n . x . n. matrix . A. is a nonzero vector . x. such that . 7.1. Eigenvalues and Eigenvectors. Def.. Let . A. be an . n. x. n. matrix and let . X. be an . n. x. 1 matrix. . X. is said to be an eigenvector for . A. if there is some scalar λ so that . AX = . . 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. and . eigenvectors. Births. Deaths. Population. . increase. Population. . increase. = . Births. – . deaths. t. Equilibrium. N: . population. . size. b: . birthrate. d: . deathrate. The. net . 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 . 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 . Mark Hasegawa-Johnson. 9/12/2017. Content. Linear transforms. Eigenvectors. Eigenvalues. Symmetric matrices. Symmetric positive definite matrices. Covariance matrices. Principal components. Linear Transforms. Review. If . . (. is a vector, . is a scalar). . is an eigenvector of A . . is an eigenvalue of A that corresponds to . . Eigenvectors corresponding to . are . nonzero. solution . of . (. A. .