solve a matrix equation

fengming

2012 6 28

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any one has any ideas about how to solve a matrix equation like (inv([sI-A])*B*w)'*E=0, where I is a unit matrix, A, B and E are known matrices, s is a scalar variable and w is a matrix variable with corresponding dimension. or how to find s and w to minimize the norm of (inv([sI-A])*B*w)'*E ?

For example, if A=[1 2;-1 2]; B=[1 2];E=[1 1];

any ideas are appreciated

Thanks

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fengming 2012-6-29

Thank you for your reply. sorry for the mistakes. the example should be A=[1 2;-1 2]; B=[1 2]';E=[1 1]'; B and E are vectors, and in this case, w is a nonzero scalar.

and another example: A=[1 2;-1 2]; B=[1 2;2 1]';E=[1 1]'; in this case, w should be a 2-by-1 nonzero vector. then sI-A is a 2-by-2 matrix; and the inv(sI-A) is also a 2-by-2 matrix; then inv(sI-A)*B would be 2-by-2, inv(sI-A)*B*w would be 2-by-1, and the transpose of inv(sI-A)*B*w is 1-by-2 and (inv(sI-A)*B*w)'*E would be a scalar.

if B is m-by-n and E is p-by-q, w would be n-by-q.

2 norm or Infinity norm are both OK, I prefer 2 norm.

Thank you!

Star Strider 2012-6-29

编辑：Star Strider 2012-6-29

It would be helpful to know where this equation comes from and the context in which the question is being asked. The equation looks suspiciously like a Laplace-transformed state equation, usually equated not to zero but to the Laplace-transformed state vector 'X'. The matrix '[s*I-A]^(-1)' is the Laplace transform of the 'fundamental solution matrix' or 'state transition matrix', the inverse transform of which is usually expressed as 'expm(A*t)' or something similar, depending on the context. In addition, 'w' might be white process noise.

In short, I am not certain this question has an answer, at least not with the equation in this form. If I am wrong, I invite correction, ideally with a detailed explanation or online reference.

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