## Description

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

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