MATLAB: Force a linsolve solution

linsolvesingular solution

I have a state space system with a Cn matrix
Cn = [1.318e+5 0 0 0;0 0 1.318e+5 0]
and I need to transform it to a C matrix like
C = [1 0 0 0; 0 0 1 0]
The transformation can be made using the following: A = M*An*inv(M), B = M*Bn, C = Cn*inv(M)
I tried to use linsolve(C,Cn) to get my M matrix, which "works" ang gives me
M =
      131800           0           0           0
           0           0           0           0
           0           0      131800           0
           0           0           0           0
But this matrix M is non-invertible and the multiplication Cn*inv(M) returns a matrix of NaN values. I know that:
M =
        131800           0           0           0
             0           1           0           0
             0           0      131800           0
             0           0           0           1
is invertible and works perfectly for me in this case, but I can't find a way to calculate this M on Matlab.
Can I force the result of a linsolve to be invertible or is there another way to calculate this M so it gives me the one I desire?

Best Answer

NO. You cannot force linsolve to return the solution you wish to see for a non-invertible matrix. At least not without doing some moderate linear algebra of your own to reduce the dimension of the problem. But if you knew how to do that, you would not be asking how to use linsolve here.
You can use a tool that is designed to survive on singular problems.
pinv(Cn)*Cn
ans =
   1     0     0     0
   0     0     0     0
   0     0     1     0
   0     0     0     0
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