Topic 1: Matrix functions with Chebyshev polynomials Let A RV*Y be a symmetric matrix. In the homework we have explained how to form polyno- mials of matrices p(A). More general functions f(A) of matrices can be constructed as limits of polynomials. By approximating general functions f with Chebyshev polynomials, one can derive efficient algorithms for performing matvecs by f(A) using only matvecs by A as the main ingredient. In this project, we will explore how this can be accomplished. 1. By the spectral theorem, A can be factorized as A = UAU ' where U is orthogonal and A is diagonal. In terms of U and A, deduce the diagonalization of p(A) for a given polynomial p. 2. Suppose that the eigenvalues of A are contained within the interval [a,b]. Suppose moreover that f is a real-valued function whose domain contains [a, b] and that p;, k = 0,1,2, ..., define a sequence of polynomials that converge pointwise to f on [a,b]. Define (4) = lim py(4). In terms of U and A, deduce the diagonalization of f(A). Explain why the definition of f(A) does not depend on the specific choice of sequence {px} but only on the limiting function f. 3. Explain why if [a, b] does not contain 0, then by taking the function f(z) = 1, f(A) coincides with A~1. Moreover, for fl@) =) ax(zo)* k=0 that is given by a convergent Taylor series expansion (such as, for example, f(z) = e*), explain why o0 FA) =) an(Acl)E. k=0