Unsolvable Linear Algebra System: Need Help with Least Squares Method

[WCMU101]

Hey all. I'm not too sharp on linear algebra. I've done a first year university course on it, but that was a couple years ago & didn't go into much detail. Here is the problem:

Matrix I (n,1).
Matrix N (m,1).
Matrix G (n,m).

Now... I is a column vector of computed "double differences". These double differences are composed of original elements N. Matrix G relates the original elements to the double differences, such that:

G*N = I

G is composed of 1s and -1s, here is an example: (See attachment). So using that G:

n = 3, m = 8

So I already know the system can not be solved, since there are 8 unknowns (N) and only 3 equations. Hence I decided I needed more observations & I'll use least squares to get a solution. So for example:

[G G G]^T*N = [I1 I2 I3]^T

Where G is the same as it was above. And I1,2,3 correspond to the observations at time 1, 2 and 3 respectively. N is still 8x1.

So by least squares:

Let [G G G]^T = G_Big

G_Big^TG_Big*N = G_Big^T[I1 I2 I3]^T
N = (G_Big^TG_Big)^-1G_Big^T[I1 I2 I3]^T

Problem is (G_Big^TG_Big)^-1 is singular, I can't take the inverse. It happens for all sizes of G.

I don't know how else to solve this system.

Any help would be greatly appreciated!

Thanks. Nick.

 

[hotvette]

From a mathematical standpoint, singular systems have either no solution or an infinite number of solutions. In the latter case, there is something called a mininum norm solution, but that probably won't be very helpful in a physical experimentation based scenario (i.e. a mathematical solution won't likely represent the physical phenomenon).

Can you tell us more about the physical situation that's being modeled? If you are taking data points and have 8 variables, you need to be able to have data points with all 8 unknowns being varied. Looks like you are collecting data w/o some of the variables changing.