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Suppose that a vector $x$ is given. We want to find out the total number of possible ways to form a matrix $A$ so that $Ax = Ix$ where $I$ is $n \times n$ identity matrix and $A$ is some $n \times n$ matrix.

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    That matrix will not be unique. Notice that this simply means that $A$ has $x$ as eigenvector with 1 as corresponding eigenvalues. the remaining elements could be arbitrary.2012-12-11
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    I edited the question.2012-12-11

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