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Main Question or Discussion Point
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^{T}G_Big*N = G_Big^{T}[I1 I2 I3]^{T}
N = (G_Big^{T}G_Big)^{1}G_Big^{T}[I1 I2 I3]^{T}
Problem is (G_Big^{T}G_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.
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^{T}G_Big*N = G_Big^{T}[I1 I2 I3]^{T}
N = (G_Big^{T}G_Big)^{1}G_Big^{T}[I1 I2 I3]^{T}
Problem is (G_Big^{T}G_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.
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