I am not sure how to calculate this norm?

Solution 1:

You are looking at the induced 2-norm of a matrix. Induced 2-norm of a matrix is given by \begin{align} ||A||_2=\max_{x\neq 0}~\frac{||Ax||_2}{||x||_2} \end{align}
There is a bit of theory behind it which will help you derive that induced 2-norm is infact the highest singular value of that matrix. To find the highest singular value, find $AA^T$ and find the highest eigenvalue of that matrix and take its square root. The condition number is nothing but the product of induced 2-norm of $A$ and its inverse. You can find all this stuff in any standard textbook on matrix analysis.

Solution 2:

Here is how you find the norm of a matrix. Apply the definition of the norm of a matrix

\begin{align} ||A||_2 = \max_{||u||= 1}~||Au||_2. \end{align}

to the matrix you have been given. First, let's find $ ||Au||_2 $. Pick up an arbitrary vector $u=(x,y,z)^{T}$ such that $||u||_2 = 1$ and apply the given matrix to it

$$ Au= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}= \begin{bmatrix} x \\ y+z \\ z \\ \end{bmatrix} $$

$$ \implies ||Au||_2 = \sqrt{x^2+(y+z)^2+z^2} = \sqrt{(x^2 + y^2 + z^2) + 2yz + z^2}$$

$$ \sqrt{1 + 2yz + z^2}.$$

Now, we have

$$ ||Au||_2 = \sqrt{1 + 2yz + z^2} \implies ||A||= \max_{||u||_2=1 }~\sqrt{1 + 2yz + z^2}= \frac{1}{2}+\frac{\sqrt{5}}{2} . $$