Feature extraction from matrices with unequal size The goal is to do supervised learning, we have a set of inputs and target variables. But for each entry, the input is not a 1-dimensional vector. But rather there is a symmetrical 2-D matrix for each entry. Also size of the matrix for one entry is different from other ones. What are the ways to unify all info from matrices into a standard form for all entries and preserve all the information at the same time?
I know one way is to extract eigenvalues for each matrix. Is that going to preserve all information? Even then, number of eigenvalues are different for each matrix depending on the size.

 A: In my experience, problems where the input data has different sizes are usually multiple instance learning (MIL) problems in disguise. (wikipedia has 2 different entries for MIL: https://en.wikipedia.org/wiki/Multiple-instance_learning and https://en.wikipedia.org/wiki/Multiple_instance_learning)
In my opinion, the best paper to learn about MIL is Amores, Jaume (2013), "Multiple instance classification: Review, taxonomy and comparative study", Artificial Intelligence, 201: 81–105, 
In a nutshell, in MIL you have to generate a different representation of the data. Historically the first solution was to generate features by aggregating the data in the original representation - taking mean, standard deviation, maximum and minimum values, and so on. Thus the data may have different sizes but you end up representing the data using this fixed set of aggregation measures. 
The second family of approaches are vocabulary based. In some way  you learn "frequent sub-patterns of the data" ( if your data is an image, the sub-patterns can be patches of the image, if the data is a time series, they can be segments) and you represent (in very general terms) the data as the histogram of such sub-patterns for each of the original data. Again, the data has different sizes but the set of sub-patterns is fixed, so the histograms (number or frequency of each sub pattern appears in the data) are of fixed size (sometimes this vocabulary approach is called "bag of visual word" when the data are images).
Please read Amores papers. I learned a lot from it. 
My experience in MIL was mainly in images and for a long time I used bag of visual words. In a time series problem, to my surprise, the aggregation approach was the best solution. Now, everybody and me also, are using deep nets to solve MIL problems!!
A: For machine learning and statistical prediction problems, it is unlikely that there will be any value in trying to coerce symmetric matrices of different sizes to a uniform size.  This will generally either involve extending small matrices to large ones (thus having many useless entries), or coercing larger matrices to smaller ones (which is messy and unlikely to illuminate your problem).
Technically it is possible to map real matrices of any (finite) size down to real numbers, and vice versa.  Any $n \times n$ real matrix (for finite $n$) is cardinally equivalent to a single real number, so you can map either way.  There are innumerable ways that you could coerce these to equivalent size, either by extending the smaller ones (e.g., zero-padding each matrix to the size of the largest matrix), or by doing a nasty conversion down to smaller size (e.g., collapsing multiple real numbers to a single real number using annoying strings of alternating digits from different numbers).
Without knowing the context of your problem, it is difficult to say what is best here, but it is very unlikely that any of these methods is going to be helpful.  Extending smaller matrices to bigger ones without any additional information is essentially just a waste of extra matrix entries, and coercing larger matrices down to smaller ones is messy and does not save information.
Perhaps a better idea is to take a step back from your problem and think about the formulation of the underlying probabilistic mechanism generating these observable matrices.  If your statistical problem involves observing symmetric matrices then your model ought to be written in a way that takes a vector of these matrices, allowing for different sizes.
