How to solve A*X = B*Y

Hello,

I would like to solve the following equation : A*X = B*Y where A and B are n-by-n known matrices. X and Y are vectors of size n. The first k elements of X and the last n-k elements of Y are known. Does anyone have a clue ?

Thank you very much for your help!

[Paul]

One way to resolve this problem is to solve the first k equations of the system X = A\B*Y : it a system with k equations and k unknowns but I don't know how to solve it with Matlab.

Answer 1

[Matt J]

If you partition X into known and unknown parts

 X=[x1 ; x2]

and similarly

Y=[y1;y2]

then the equation can be rewritten

A(:,k+1:end)*x2 - B(:,1:n-k-1)*y1=   B(:,n+1-k:n)*y2 -  A(:,1:k)*x1

The RHS of the above consists of all known quantities and reduces to some known vector b. The LHS can be combined into a single linear system which you can solve using normal methods.

[Paul]

Thanks a lot Matt !

Answer 2

[Jan]

== This is not valid Matlab syntax ==

The first k elements of X are known and the last n-k of Y:

X = [q; x] = [q; 0] + [0; x]   % Here "0" means a zero vector
Y = [y; p] = [y; 0] + [0; p]

Now you get:

A*X = A*[q; 0] + A*[0; x] = B*[y; 0] + B*[0; p]
A*[0; x] - B*[y; 0] = B*[0; p] - A*[q; 0] = C
[A(:, 1:k), B(:, k+1:n)] * [y; x] = D * [y; x] = D*z = C

z = D\C;

X_unknown = z(k+1:n);
Y_unknown = z(1:k);

[Paul]

Thanks a lot Jan !