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I have tow matrix A & B, that B is a parametric matrix. what i can find B so that it is commuting with A?

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closed as not a real question by Bill Johnson, Chris Godsil, Andreas Blass, Suvrit, Andrés Caicedo Jul 1 '12 at 3:36

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

2 Answers 2

The question is a bit vague, but in some cases it can certainly be clarified. If A is a general square complex matrix, then to commute with A is to be a polynomial in A. This follows because A can be diagonalised with distinct eigenvalues. This is a case in which the matrices commuting with A have the smallest possible dimension.

Your parametric matrix B is geometrically a curve in matrix space (assuming one parameter). The only reason to believe that the curve must intersect the commuting space of A would be dimension. In general there will be no reason for an intersection. Therefore you need more parameters in B; or you need A to be more special; or you need the size of the matrices to be very small (e.g. 1 or 2).

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If your question is about finding the values of parameters such that $A$ and $B$ commute, just solve the system of $n^2$ (if the matrices are $n \times n$) equations in the parameters that say each entry of $AB-BA$ is $0$.

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