Syllabus for MATH125 Nicolas Guay

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?)

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-

. A is an n x n matrix. Mark each statement True or False. Justify each answer. i. If Ax ?? for some vectors, then ? is an eigenvalue of A ii. A matrix A is not invertible if and only if 0 is an eigenvalue of A iii. A number c is an eigenvalue of A if and only if the equation (A-c)x = 0 has a nontrivial solution. iv. Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy v. To find the eigenvalues of A, reduce A to echelon form.