Background:
I am an engineer and sometimes I measure or estimate things like this
$$ {\bf T} = \sum c_k{\bf v_k}{\bf v_k}^T \cases{ c_k \in \mathbb R^+\\ {\bf v_k} \in \mathbb R^{N\times 1}}$$
In other words ${\bf T} \in \mathbb R^{N \times N}$ so that $\bf T$ is a positive semidefinite form.
Normalization is a useful thing to be able to do, and sometimes Frobenius norm is useful, but in other cases we may do better with spectral norm.
The spectral norm is defined as the largest singular value: $\sigma_1 > \sigma_2 > \cdots > \sigma_N$ if $${\bf T}={\bf U\Sigma V}^T$$Usually sorted like this in a diagonal matrix : $$ {\bf \Sigma} = \begin{bmatrix}\sigma_1&&\\&\ddots&\\&&\sigma_N\end{bmatrix}$$
Where as our $\bf T$ also is guaranteed to be symmetric, which means we can assume $\bf V=U$
Now to the question, how can I normalize a given $\bf T$ with it's largest singular value, without having to calculate anywhere near a full SVD so that I can get $${\bf \hat T = V} \begin{bmatrix} 1&&&\\&\sigma_2/\sigma_1&&\\&&\ddots\\&&&\sigma_N/\sigma_1 \end{bmatrix} {\bf V}^T $$
Own work : What I need as far as I can see is to calculate $\sigma_1$ which I know a rather primitive method for doing, namely the "power method". Taking a random vector $\bf r_0$, iterating the following :
$${\bf r}_{n+1} = {\bf T r}_n$$ And after a number of iterations we shall have an estimate of $\sigma_1$ by looking at the element-wise division. ${\bf r}_{k} / {\bf r}_{k-1}$ as the portion of r growing fastest shall be in the direction of the singular vector corresponding to $\sigma_1$