I am not sure how to calculate this norm? I have the following matrix: $$A=
        \begin{bmatrix}
        1 & 0 & 0 \\
        0 & 1 & 1 \\
        0 & 0 & 1 \\
        \end{bmatrix}
$$
What is the norm of $A$? I need to show the steps, should not use Matlab...
I know that the answer is $\sqrt{\sqrt{5}/2+3/2}$. I am using the simple version to calculate the norm but getting different answer: $\sum_{i=0}^3\sum_{j=0}^3(a_{ij})^2=\sqrt{1+1+1+1}=2$
Maybe this is some different kind of norm, not sure.
This might help - i need to get a condition number of $A$, which is $k(A)=\|A\|\|A^{-1}\|$...that is why i need to calculate the norm of $A$. 
 A: 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. 
A: 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
$$ ||Au||_2  = \sqrt{1 + 2yz + z^2} \implies ||A||= \max_{||u||_2=1 }~\sqrt{1 + 2yz + z^2}= \frac{1}{2}+\frac{\sqrt{5}}{2} . $$
