In this problem, you will find the eigenvalues of matrix in Problem where 0n Try solutions of the form jand show that are distinct eigenvalues of Use trigonometric identities including 2 . We...

In this problem, you will find the eigenvalues of matrix
in Problem

where

<sub>0

n</sub>

Try solutions of the form

_j
and show that

are distinct eigenvalues of

Use trigonometric identities including

²

.

We know that

is positive definite. Are its eigenvalues consistent with that

Find the condition number of
and show that the matrix is ill-conditioned for large

Remark
With homogeneous boundary conditions

it can be shown due to the form of the right-hand side

that

where

is a constant independent of

In short, the system

can be solved reliably (see Ref.