Way to solve AX=XB

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SATISH SONWANE
SATISH SONWANE 2023-1-25
编辑： Bruno Luong 2023-1-28
Is there any implementation of Tsai and lenz's (Or any other) method for solving AX=XB for hand- Eye Calibration?

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回答（3 个）

the cyclist
the cyclist 2023-1-25
This is a special case of the Sylvester equation.
Looks like the sylvester function will be helpful for you.
You might also be interested in this blog post on the topic by Cleve Moler.
Torsten
Torsten 2023-1-28
编辑：Torsten 2023-1-28
dim = 4;
X = sym('X',[dim dim]);
A = rand(dim);
B = A.';
[M, ~] = equationsToMatrix(A*X==X*B)
if rank(M) < size(A,1)^2
N = null(M);
for i = 1:size(N,2)
S{i} = reshape(N(:,i),size(X));
S{i}
A*S{i}-S{i}*B
end
end
Matt J
Matt J 2023-1-28
编辑：Matt J 2023-1-28
[ma,na]=size(A);
[mb,nb]=size(B);
%size(X)=[na,mb]
X=null( kron(speye(mb),A) - kron(B.',speye(na)) );
X=reshape(X,na,mb,[]);
  2 个评论
the cyclist
the cyclist 2023-1-28
Ran in:
I couldn't get this method to work. Am I overlooking something dumb?
rng default
A = rand(5);
B = rand(5);
[ma,na]=size(A);
[mb,nb]=size(B);
X=null( kron(speye(mb),A) - kron(B.',speye(na)) );
Error using svd
SVD does not support sparse matrices. Use SVDS to compute a subset of the singular values and vectors of a sparse matrix.

Error in null (line 75)
[V, s] = svd(A','vector');
X=reshape(X,na,mb,[]);
Bruno Luong
Bruno Luong 2023-1-28
编辑：Bruno Luong 2023-1-28
Ran in:
null can only work wth full matrix
rng default
A = rand(5);
XX = rand(5);
B = XX\(A*XX);
[ma,na]=size(A);
[mb,nb]=size(B);
K=null( kron(eye(mb),A) - kron(B.',eye(na)));
R = rand(size(K,2),1); % Any random vector with this size will do the job
X = reshape(K*R,[na,mb])
X = 5×5
0.1100 0.1626 0.0772 -0.0458 0.0810 -0.0536 -0.3884 0.0510 0.0552 -0.0602 0.1937 -0.1993 0.3240 0.3508 -0.0352 -0.1165 0.4388 -0.3413 -0.3089 0.1851 0.0500 0.0277 0.0913 0.1389 0.0450
norm(A*X-X*B)
ans = 7.4501e-16

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