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Given a matrix $Y\in M_n(\mathbb{R})$, consider the set $C(Y)=\{X\in M_n(\mathbb{R}):YX=XY\}$. I was asked to show that $C(Y)$ is a vector space. I did this by showing that $C(Y) $ is a subspace of $M_n(\mathbb{R})$. ( I think this is what I was supposed to do, but feel free to point out if this does not prove/is not enough to prove what I was asked)

Now, consider the linear transformations $I_Y, D_Y: M_n(\mathbb{R})\to M_n(\mathbb{R})$, determined by $K \mapsto YK, K\mapsto KY$, respectively. My question here is: is it possible to determine $C(Y)$ from $I_Y $ and $D_Y$? Or perhaps their matrices?

user926356
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  • What do you mean by "determine"? – Gerry Myerson Sep 18 '23 at 07:25
  • Is it enough? This depends on the homework. Is it this problem? – Dietrich Burde Sep 18 '23 at 08:16
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    $C(Y)$ is, by definition, the kernel of $I_Y - D_Y$. (This shows immediately that it's a subspace.) – Qiaochu Yuan Sep 18 '23 at 14:51
  • @DietrichBurde Well, I was given the matrix $Y$ and the definition of $C(Y)$, as well as the previously mentioned linear transformations $I_Y,D_Y$. I was asked to prove that $C(Y)$ is a vector space, which I've done. But my main concern is how to define $C(Y)$ using such linear transformations (or their matrices) – user926356 Sep 18 '23 at 17:23
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    Your "main concern" is solved, as Qiaochu said above. Your homework might go on, to show that its dimension is at least $n$? Or why do you need that $C(Y)$ is a subspace? – Dietrich Burde Sep 18 '23 at 18:43

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