Preparation guide for the final Definitions and notions to know =============================== Spectral radius of the matrix Convergent matrix Positive definite matrix A-orthogonality of vectors Orthogonal matrices Similar matrics; similarity transformations Power method (for finding an eigenvector that corresponds to the largest eigenvalue) Inverse power method; inverse power method with a shift Householder reflector (transformation) Upper Hessenberg form Things to know without a proof ============================== Explain why for each eigenvalue there is at least one eigenvector Thm 7.17 General fixed point iterations Jacobi iterations Gauss-Seidel iterations Give an example of a 3 by 3 matrix A such that the straightforward fixed point iterations (with the matrix I-A) will not converge, but Jacobi iterations will. What orthogonality conditions are satisfied by the directions (vj) and by the residuals (rj) in the conjugate gradient method Thms 9.9, 9.10 Given vectors x and y, build a reflector (find the corresponing matrix A) that reflects x into y (i.e. such that A is a reflector, and Ax=y). Forward and backward difference schemes for the simplest heat equation. Things that require derivations or proofs ========================================= Give an example of 3 by 3 matrics that has only one (linearly independent) eigenvector. Explain rigorously why there is only one eigenvector. Thm 7.15 (part II only) Prove that if some induced (natural) matrix norm is less than one then the spectral radius < 1. (Use the result of theorem 7.17). Lemma 7.18 Theorem 7.19 (The first part only, i.e. spectral radius < 1 implies convergence of the fixed point iterations) Corollary 7.20 (with the proof, see notes) Thm 7.21 (prove for the Jacobi iterations) Thm 7.31 (the proof in the notes is simpler, recommended) Given vectors x and v, find t0 that minimizes g(x) =-2, i.e. such t0 that g(x+t0*v) < g(x+t*v) for any t not equal to t0. Thm 9.8 Using the result of Thm 9.10, prove Corollary 9.11 (the proof in the notes is simpler) Thm 9.12 Explain why the inverse power method with the shift mu will converge to the eigenvalue closest to mu. Thm 9.17 Explain (in detail) how to use Householder reflectors to reduce a matrix to the upper Hessenberg form.