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Let $w:\mathbb{R} \times \mathbb{R}^n \rightarrow Mat(n,\mathbb{R})$ be a smooth function, $R_{ij}$ be a fixed skew-symmetric $n\times n$ real matrix, and $A\in\mathbb{R}$. Consider the equation $$\frac{\partial w}{\partial t} - \sum_{i=1}^n\left(\frac{\partial}{\partial x^i} + \frac{1}{4}\sum_j {R}_{ij} x^j\right)^2 w + Aw = 0.$$

My question is, if one wants to solve for $w$, why is it sufficient to take $R_{ij}$ to be block-diagonal, consisting of $2\times 2$ skew-symmetric blocks? I know that a skew-symmetric matrix can always be block-diagonalised this way, but am having a hard time writing down rigorously what the solution to the equation would be if $R$ was changed to a block-diagonal form, say $P^{-1} RP$.

I'm looking for a proof that shows something like, if $w$ solves the above equation, then $P^{-1}wP$ solves the equation with $R$ replaced by $P^{-1}RP$.

(This question arose from trying to understand the proof of Proposition 12.25, p. 162, of John Roe's book on the Atiyah-Singer index theorem. http://www.maths.ed.ac.uk/~aar/papers/roeindex.pdf)

Thanks for any help.

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    $\begingroup$ Does it even matter that $w$ is a matrix? If I understand the notation correctly, the $\sum (\partial_i + \frac14 \sum R_{ij} x^j)^2$ is a scalar operator, equal to $$ \triangle + \sum_{ij} \frac12 R_{ij} x^j \partial_i w + \frac1{16} (x^T R^T R x) $$ $\endgroup$ Aug 2, 2016 at 0:36
  • $\begingroup$ Yes I think you're right. I can see that the term $\frac{1}{16}(x^T R^T Rx)$ is invariant under change of basis for $x$, but what about the other two? $\endgroup$
    – geometricK
    Aug 2, 2016 at 4:59
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    $\begingroup$ For orthogonal changes of bases, obviously yes. (A simple calculus exercise). $\endgroup$ Aug 2, 2016 at 13:01
  • $\begingroup$ Ah I see, you use orthogonality. Although it seems the only way to see what happens when you transform the middle term is to do some long calculations (which of course is easier to verify if one knew it to begin with...) Thanks for your help. $\endgroup$
    – geometricK
    Aug 2, 2016 at 15:20

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