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Homework 5 Problem 1 Show that vk is perpendicular to vk+1 in the steepest descent method. Problem 2 Let A be an n × n matrix, not assumed to be symmetric or positive definite. Assume the existence of an A-orthonormal system {u1 , . . . , un } (note that this means that U T AU = I, where U is n × n matrix whose columns are u1 , . . . , un ). Show that A is symmetric and positive definite. Problem 3 Devise a simple modification of power method to handle the following case: λ1 = −λ2 > |λ3 | ≥ . . . |λn |. Problem 4 The design of a mechanical component requires the maximum principal stress to be less than the material strength. For a component subjected to arbitrary loads, the principal stresses (σ) are given by the solution of the equations Ae = σe, where σx σxy σxz A = σxy σy σyz . σxz σyz σz Here σx , σy and σz denote the normal stresses acting along x, y and z directions, σxy , σxz and σyz denote the shear stresses acting in the xy, yz and xz planes respectively, and e = (ex , ey , ez ) represents the unit vector. Determine the largest principal stress and principal direction in a machine component for the following stress condition (in MPa) using power method. 10 4 −6 A = 4 −6 8 . −6 8 14 1