Sorry for the vague topic, but I think my question can be better explained using the following problem I came across.
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Let $A$ be a complex matrix with distinct eigenvalues define $S_{A}$ such that $S_{A}(B)=AB-BA$ (Where $B$ is a nn matrix), Then what are the null spaces and the ranges of $S_{A}$ Is it diagonalizable? And what are the eigen values, and eigen vectors.
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So here everything is pretty straightforward if we assume $A$ is diagonal.(I think the standard basic happens to be eigen vectors) But I am not sure how to proceed in the case it is not diagonal. And even in general I have seen a lot of such problems where everything is easy for a diagonal matrix. Is there a standard way to deal with these problems? Do we lose generality if we assume A is diagonal?
I again apologize if I couldn't phrase this properly but its something which I had in my mind for a while
Thank You
No, there is no loss of generality, at least as far as this specific problem is concerned. Let $P$ be a $n\times n$ invertible matrix such that $PAP^{-1}$ is a diagonal matrix $D$. Then\begin{align}S_A(P^{-1}BP)&=AP^{-1}BP-P^{-1}BPA\\&=P^{-1}DPP^{-1}BP-P^{-1}BPP^{-1}DP\\&=P^{-1}(DB-BD)P\\&=P^{-1}S_D(B)P.\end{align}So, $S_A$ and $S_D$ have basically the same properties.