Solving under-determined matrix equations

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Subhra 2013-2-18

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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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Youssef Khmou 2013-2-18

hi, are you sure that x,y,c and are nx1? just to make sure

Subhra 2013-2-19

You are right. c and d are m x 1. only x and y are n x 1.

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Youssef Khmou 2013-2-19

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编辑：Youssef Khmou 2013-2-19

在 MATLAB Online 中打开

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);