In engineering sometimes a limit of a $p$-norm, (or in practice for some conveniently large $p$):
$$\underset{x_k \text{ s.t. }x_k>x_j \forall j}{\underbrace{{\max({\bf x})}}} \approx \underset{p\to \infty}{\lim}\|{\bf x}\|_p=\underset{p\to \infty}{\lim}\left(\sqrt[p]{\sum_{\forall i} ({x_i})^p}\right)$$
is used as a continous approximation to the "max"-function, finding an approximation to the largest scalar $x_k$ in $\bf x$. Could we find some natural way to extend this idea to matrices? I was thinking to maximize eigensystem in some sense. For illustration positive semi-definite matrices correspond to ellipsoids, I would want the ellipsoid to be the maximum "radius" in every spectral dimension. Maybe in some sense the smallest ellipse still able to contain all the other ellipses (each $x_k$ being a matrix).
How can I do that (assuming we can define a branch of root in some suitable sense), would it be to just plug and chug using matrix exponentiation and then performing that root?
EDIT : if using octave matrix exponentiation (which I don't know exactly how it does this)
M^(1/k)
first having summed in a for loop:
M = M + T(:,:,n)^k;
$\bf T$s are ${\bf T}_n = {{\bf R}_n}^T{\bf R}_n$, with ${\bf R}_n \in {\mathbb R}^{2\times 2}$ and with elements of ${\bf R}_n$ sampled from normal distribution. And using $k=32$ we plot the ellipsoids (the images of equally spaced vectors on a circle):
so in some sense this could be doing what we want. Still open for new ideas!