Solving linear system of equations with random coefficients

Question

Hello,

I am a trying to see if anyone could point me in the right direction for solving a linear system of equations where the coefficients come from a normal distribution, but each equation's normal distribution has different parameters.

As a follow up, curious if there is a way to handle this problem when you have an overdetermined system but you know there are only d "unique" parameters sets for the normal distributions (where d is the number of unknowns). In other words, in our overdetermined system we essentially have sets of duplicate equations as the coefficients come from the same distribution. Is there a way to reduce those duplicates and then solve it using techniques from the first question (if that is even possible)?   

Hoping I am properly conveying my questions, I don't necessarily have the strongest math background. Appreciate any help or pointers!

Answer 1

What sorts of questions are you trying to answer?

Obviously you can only hope to say something about the *distribution* of solutions Ax = 0 is A if a random matrix.

But if you are simply given a linear system, there are myriad effective numerical techniques to efficiently compute solutions. So are you just looking for efficient numerical techniques given that A has such a structure?

Or is it something deeper and statistical you want to say about the solutions?

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>In other words, in our overdetermined system we essentially have sets of duplicate equations as the coefficients come from the same distribution.

*Duplicate* isn't that meaningful of a word in this context. Perhaps a system where an equation is  *linearly dependent* on the other equations is more precise. But just because you're generating coefficients from the same distribution doesn't mean they equations will be linearly dependent.

For example if the coefficient distribution is just a simple binomial {0, 1},  then we can obviously get systems of equations like  x = 0 and y = 0, which has a unique solution, or 0 = 0, which is underdetermined and has infinitely many solutions.