Perfect thermal contact

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System of bilinear equations look like the following for for some integer where are matrices and are some real numbers. These arise in many subjects like engineering, biology, statistics etc.

Solving in integers

We consider here the solution theory for bilinear equations in integers. Let the given system of bilinear equation be

This system can be written as

Once we solve this linear system of equations then by using rank factorization below, we can get a solution for the given bilinear system.

Now we solve first equation by using smith normal form, given any matrix , we can get two matrices and in and , respectively such that , where is as follows:

where and for . It is immediate to note that given a system , we can rewrite it as , where and . Solving is easier as the matrix is somewhat diagonal. Since we are multiplying with some nonsingular matrices we have the two system of equations to be equivalent in the sense that the solutions of one system have one to one correspondence with the solutions of another system. We solve , and take . Let the solution of is

where are free integers and these are all solutions of . So, any solution of is . Let be given by

Then is

We want matrix to have rank 1 so that the factorization given in second equation can be done. Solving quadratic equations in 2 variables in integers will give us the solutions for a bilinear systems. This method can be extended to any dimension, but at higher dimension solutions become more complicated. This algorithm can be applied in Sage or Matlab to get to the equations at end.

References