First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to do. I'll update this post to better target a particular suggestion if one is more popular than any others.
I have a matrix $A \in \mathbb{R}^{n\times m}$. I am looking for some single value or vector $\Bbb{v}\in\mathbb{R}^c$, $c$ constant, that captures a 'good amount' of information about the values in the matrix.
My very elementary understanding of the determinant is that it somewhat provides this for a $p\times p$ matrix, but of course is undefined for a non-square matrix like my $A$.
I have considered some combination of the minor determinants - taking $|A_0|$ for $A_0 \in \mathbb{R}^\phi$, where $\phi$ is the smaller of $n, m$. Followed by $|A_1|$ for $A_1 \in \mathbb{R}^{m+n-2\phi}$, and so on until I either run out of minors, or (I suppose what is the same) reach minors that are themselves single elements of $A$.
Before considering what linear combination of these $A_k$ would be suitable, I am more concerned with how exactly I should compute the determinant - or even if it is the best technique. Allow me to explain my motivation:
I would like to use this 'representative value' (or vector $\Bbb{v}\in\mathbb{R}^c$, $c$ constant) to determine if there has been some significant change between observations of $A$.
I am restricted in that I observe each element $a_{ij}$ independently, with $j$ varying more rapidly than $i$. Thus any technique that minimises the number of variables 'stored' (that is, some $z=\sum_{x=0}^{y}f(x)$ while $x<y$, for example) is a bonus. Just as the final result, the number of such $z$ must be independent of $n,m$.
1) Is some combination of $A_k$ a good/valid method?
2) What would be a good way to combine them?
3) What would be a better method?