I have tow matrix A & B, that B is a parametric matrix. what i can find B so that it is commuting with A?
2 Answers
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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).
answered Jun 29, 2012 at 11:35
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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 $0$.
answered Jun 29, 2012 at 17:38