1.

Let

Compute, by hand, the eigenvalues and eigenvectors of A, and use them to solve Ax = b for x (use the "spectral method").

2.

Repeat Exercise 1 for

3.

Repeat Exercise 1 for

4.

Repeat Exercise 1 for

5.

Let A ? R ^{n n} be symmetric, and suppose the eigenvalues and (orthonormal) eigenvectors of A are already known. How many arithmetic operations are required to solve Ax = b using the spectral method?

6.

A symmetric matrix A ? R ^{n n} is called positive definite if

Use the spectral theorem to show that A is positive definite if and only if all of the eigenvalues of A are positive.

7.

Let L be the n n matrix defined by the condition that

where h = 1/( n + 1). For example, with n = 5,

1. For each j = 1,2, , n, define the discrete sine wave s ^{( j}) of frequency j by

   

   Show that s ^{( j)} is an eigenvector of L, and find the corresponding eigenvalue ? _j. (Hint: Compute Ls ^{( j)} and apply the addition formula for the sine function.)

2. What is the relationship between the frequency j and the magnitude of ? _j?

3. The discrete sine waves are orthogonal (since they are the eigenvectors of a symmetric matrix...