Hint for a problem on condition number

[kalish1]

I would like to know if the second part of this question is asking something different.

**Problem:** Consider the linear system $19x_1+20x_2=b_1, 20x_1+21x_2=b_2$. Compute the condition number of the coefficient matrix. Is the system well-conditioned with respect to perturbations of the right-handside constants ${b_1,b_2}$?

Do I need to introduce a $\delta$ into the right-handside, or is computing the coefficient number enough to conjecture about the condition of the right-handside constants?

Thanks.

 

[Ackbach]

The second part of the problem as asking if the condition number you just computed is highly dependent on $b_{1}$ and $b_{2}$. What happens if you change the RHS's just a little? Does the condition number change a lot when you do that? I don't think you need to introduce another variable, at least not yet. What do you get for the condition number? And how are you computing it?

 

[kalish1]

I get 1601 = 41*41 for the condition number, and I got it by computing the norm of the matrix A and the norm of the matrix A^(-1), an then multiplying them together. The norms of the matrices are both 41.

Isn't this enough for the purposes of this problem?

 

[Ackbach]

Did you use the formula
$$\|A\|= \max \{ \|Ax \|:x \in \mathbb{R}^{2}, \|x \|=1 \}?$$
If so, what vector norm did you use? Euclidean?

 

[kalish1]

I used the following norm:

$$\|A_{n\mathbb x n}|=\max_{1\leq i \leq n}\sum_{j=1}^{n}\|a_{ij}\|$$