TITLE: How do I calculate the $p$-norm of a matrix?
QUESTION [11 upvotes]: I know that the $p$-norm for a matrix is:
$$\|A\| = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}$$
but I don't know what this really means.
So how would I compute the $2$-norm, $3$-norm, etc for the matrix.
$$A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$$
UPDATE
Apparently, the above matrix is too easy :) Let's try something harder.
$$A = \begin{bmatrix} 2 & 1 & 4 \\ 3 & 0 & -1 \\ 1 & 1 & 2 \end{bmatrix}$$
Thanks,

REPLY [3 votes]: For other values of $p$, you can use Lagrange multiplier methods.  Here, for example, is your $3 \times 3$ example with $p=4$ using Maple. 
> M:= <<2|1|4>,<3|0|-1>,<1|1|2>>;
  X:= <x1,x2,x3>;
  MX:= M.X;
  F:= add(MX[i]^4,i=1..3)-lambda*(add(X[i]^4,i=1..3)-1);
  eqs:= {diff(F,x1),diff(F,x2),diff(F,x3),diff(F,lambda)};
  S:=RootFinding[Isolate](eqs,[x1,x2,x3,lambda]);
  pnorm:= max(map(t -> eval(lambda,t),S))^(1/4);

pnorm := 5.695759124