System of bilinear equations

In algebra, systems of bilinear equations are collections of equations, each one of which is written as a bilinear form, for which a common solution is sought. Given one set of variables represented as a vector "x", and another represented by a vector "y", then a system of bilinear equations for "x" and "y" can be written formula_1. Here, "i" is an integer whose value ranges from 1 to some upper bound "r", the formula_2 are matrices and formula_3 are some real numbers. Systems of bilinear equations arise in many subjects including engineering, biology, and statistics.
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 the first equation by using the Smith normal form. Given any formula_7 matrix formula_8, we can get two matrices formula_9 and formula_10 in formula_11 and formula_12, respectively such that formula_13, where formula_14 is as follows:
where formula_16 and formula_17 for formula_18. Given a system formula_19, we can rewrite it as formula_20, where formula_21 and formula_22. Solving formula_20 is easier as the matrix formula_14 is somewhat diagonal. Since we are multiplying with some nonsingular matrices, the two systems of equations are equivalent in the sense that the solutions of one system have a one-to-one correspondence with the solutions of another system. We solve formula_20, and take formula_26.
Let the solution of formula_20 be
where formula_29 are free integers and these are all solutions of formula_20. So, any solution of formula_19 is formula_32. Let formula_10 be given by
Then formula_35 is
We want matrix formula_37 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 system. 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.
See also.
Systems of linear equations