0
$\begingroup$

Let $A$ be an $n\times n$ diagonal matrix with characteristic polynomial $(x-c_1)^{d_1}\cdots(x-c_k)^{d_k}$, where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is $d_1^2+\cdots+d_k^2$.

  • 0
    Hint: start writing down an example of $A$, and then equations for $B$; once you've determined the dimension of the solution set, prove that it does not depend on the choice of $A$.2012-04-16
  • 0
    we already know the matrix $A$; it is diagonal matrix in which $c_1$ appears $d_1$ times and so on.If $B$ is any diagonal matrix then it commutes with $A$ but what are other matrices which commute with $A$?2012-04-16

1 Answers 1