What are some use of dense matrices in statistics? OK, I am not a statistician (Not even close). I am a High Performance Computing researcher and I wanted a few test cases for Large (Greater than 5000x5000) Dense Matrices. I had asked here and a few other places but never got any reply from a statistician. I am very much interested in trying out my codes on a statistics problem. Could you suggest an application in statistics where one needs to solve $Ax=b$ for x where $A$ is dense and square.
I would highly appreciate it if you could also give me applications where A has no structure i.e. No symmetry, No Positive-Definiteness etc. But thats not entirely necessary. A large dense matrix with a good application suffices.
I'm sorry if this question appears open or vague but I can't imagine a better place to ask this question.
 A: You might find the Java Matrix Benchmark useful.  The Matrix Market does not seem to have what you want, although it has many examples.
A: Here is large, although I'm not sure if it's dense enough for you. From http://www.grouplens.org/node/73


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A: I'm not sure the application you are seeking would make sense in a statistical context. What you're interested in is a linear regression analysis. $A\in R^{m\times n}$ is a matrix of $m$ measurements in which each row is a single measurement of $n$ variables. Two potential applications with possibly $n>5000$ come to my mind. 1) DNA microarray analysis and 2) analysis of functional MRI data. In any case it will be hard to find data sets with $m>5000$ people (measurements) in it. 
However, your requirement of $m=n$ restricts the sense of such an analysis in a principle way. After all statistics is about inferring some underlying, let's say, truth from noisy data, i.e., the statistical model implicit to your question is 
$$b=a^Tx + \epsilon$$ 
where $a$ is a single measurement, $x$ are the assumed parameters that you try to find with your analysis and $\epsilon$ is some form of noise. Now you say that $A$ needs to be invertible, i.e., has to be full rank, i.e., measurements $a$ must not repeat, i.e., you only have a single, noise corrupted observation $b$ per $a$ and that is a very bad situation to try to estimate parameters $x$, especially, in the case where the number of parameters exceeds (or is equal to) the number of measurements. Then your model simply fits the noise in the data which is called overfitting.
