ENGM 1041 Chapter Notes - Chapter 10: Xz, Demyship, Bucatini

A-B 3 -21 2 -2 2. [20 points] Given the matrix: (a) Use the characteristic equation to determine the eigenvalues of A. Then determine an eigenvector corresponding to each eigenvalue. (b) For the linear transformation: T(E) AR,graph the eigenvectors you gave in part (a) and their outputs. 2 input eigenvectors output vectors (c) Verify that the matrix A Hamilton Theorem.) satisfies its own characteristic equation. (This result is known as the Cayley-

2) (Adapted from an older edition of Peter Lax's book:) We define the cyclic shift mapping S, acting on vectors a e Cn, by S(x1,T2, . . . ,xn) = (xn , X 1 , ,Xn-1). Prove that S preserves the norm of vectors, and is therefore unitary (the name for orthogonal matrices with complex entries.) Determine the eigenvalues and eigenvectors of S, and verify that the eigen- values have absolute value 1 and that the eigenvectors are indeed orthog- onal. . Verify the following surprising fact: the expansion of a vector v in terms of the eigenvectors of S is the discrete Fourier transform of v 3) Think of Cn as the space of discrete periodic functions; i.e. identify an+1 with a1 and zo with zn. Consider the linear transformations corresponding to the discrete first (forward) and second derivatives, Find their eigenvalues and eigenvectors. (Hint: Do they commute with the transformation S of the second exercise?)

Consider the system of differential equations -73"1 + 2/3c2 where 1 and 2 are functions of t.Our goal is to find the general solution of this system a) This system can be written using matrices as X, = AX, where X is in R2 and the matrix A is A= asin(a)002 sin (a 00 αΩ Or b) Find the eigenvalues and eigenvectors of the matrix A associated to the system of linear differential equatons. List the eigenvalues separated by semicolons Eigenvalues Give an eigenvector associated to the smallest eigenvalue. Answer:

b) Find the eigenvalues and eigenvectors of the matrix A associated to the system of linear differential equatons. List the eigenvalues separated by semicolons. Eigenvalues: Give an eigenvector associated to the smallest eigenvalue. Answer: b sin(a) b sin (a Give an eigenvector associated to the largest eigenvalue. Answer:

c) The general solution of the system of linear differential equations is of the form X = c1 X1 + c2X2 where c1 and c2 are constants, and b sin (a) and ab sin (a) α Ω

ー4t We assume that X1 is assoicated to the smallest eigenvalue and X2 to the largest eigenvalue. Use the scientific calculator notation. For instance, 3e is written 3 ::.ii ะเ