Solving under-determined matrix equations
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I need to Solve : A1*x+B1*y=c; A2*x+B2*y=d in Matlab where A1, A2, B1 and B2 are m-by-n complex matrices with m<n and rank=m. x, y, c and d are n-by-1 vectors. Hence the system is under-determined. I have seen solution techniques for solving system of equations in the form Ax=b, but how can I apply that to my case? Please let me know..
Thanks in advance S.Paul
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Answers (1)
Youssef Khmou
on 19 Feb 2013
Edited: Youssef Khmou
on 19 Feb 2013
Hi Subhra,
try to check the lengths you assigned to vectors c,d ; Under-determined system means you have more variables and less equations , in your example c and d must be mx1 vectors to get under-determined system : In this case the matrix A is not square and other techniques must be used : like
1)The Moore-Penrose pseudoinverse matrix or
2)The Least squares algorithm , TLS,..etc
Try this example :
m=4;
n=6; % m<n
A1=randn(m,n)+j*randn(m,n);
B1=randn(m,n)+j*randn(m,n);
A2=randn(m,n)+j*randn(m,n);
B2=randn(m,n)+j*randn(m,n);
rank(A1) % rank(A1)=rank(A2)=rank(B1)=rank(B2)
c=rand(m,1);
d=rand(m,1);
% Matrices concatenation to get AX=B
A=[A1 B1;A2 B2]; % size{A} is 2mx2n and rank{A} is 2m .
B=[c;d]; % size{B}is 2mx1
% Solution
% Moore-Penrose pseudoinverse of matrix since A is not square :
X=pinv(A)*B;
error=X*A-B;
% LS
X2=pinv(A'*A)*A'*B;
% Now get x and y :
%x=X(1:4); y=X(5:end);
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