[SOLVED] Csci 5606, assignment 4

Problem 1, Let k · k be a norm on Rn
, and let S be an n × n nonsingular matrix. Define
kxk
0 = kSxk, and prove that k · k0
is a norm. Let k · k be a subordinate matrix norm. Show
that kAk
0 = kSAS−1k is a matrix norm.
Problem 2. Prove that if A is positive definite, then so are A2
, A3
, …, as well as A−1
, A−2
,
…
Problem 3. Program the Gauss-Seidel method and test it on the following examples



3x + y + z = 5
x + 3y − z = 3
3x + y − 5z = −1



3x + y + z = 5
3x + y − 5z = −1
x + 3y − z = 3
1