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