Let $A$ be a $n \times n $ diagonal matrix with real entries with characteristic polynomial $(x-a)^p(x-b)^q$ , where $a,b$ are distinct real numbers ; let $V:=\{B \in M(n,\mathbb R):AB=BA\}$ then $V$ is a subspace of $M(n,\mathbb R)$ , what is the dimension of $V$ ?


closed as off-topic by user26857, heropup, Shailesh, JonMark Perry, Parcly Taxel Sep 3 '16 at 3:04

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    $\begingroup$ Perhaps include your work on the problem so far? $\endgroup$ – Travis Jan 22 '16 at 11:19
  • $\begingroup$ Otherwise you might explain where the problem was found or what makes it interesting to you? $\endgroup$ – hardmath Jan 22 '16 at 12:24

The subspace you seek is the Lie algebra of the centralizer of your diagonal matrix in $GL(n)$, and has dimension $p^2 + q^2$.

Proof: Consider a matrix $B$ which commutes with diagonal matrix under consideration. Write $\mathbb R^n = V_a \oplus V_b$, where $V_a$ is the eigenspace for $a$ and $V_b$ likewise for $b$. Then for $v \in V_a$, we have $A(Bv) = B(Av) = a(Bv)$. Thus $B$ preserves the subspace $V_a$. The same argument shows it presrves $V_b$, too. Thus $B$ itself can be put in block form which consists of $p^2 + q^2$ entries.