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I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the generators always in such a way that

$$\operatorname{tr}(G^{a}G^{b})\propto \delta^{ab}.$$

How can I show this? One idea would be maybe to use something like diagonalization of the matrix with entries $\operatorname{tr}(G^{a}G^{b})$, but I don't know if this works, because the diagonal entries of this diagonalized matrix could also have different values.....Any ideas?

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic ♦
    Commented May 10, 2020 at 20:14
  • $\begingroup$ Linked. $\endgroup$
    – Cosmas Zachos
    Commented May 10, 2020 at 20:24
  • $\begingroup$ Take some a and note the hermitean $G_a^2$ is unitarily diagonalizable, with non vanishing positive trace; which you can then absorb into its normalization. You then orthogonalized all other generators w.r.t. it, suitably adjusting the r.h.s. of the Lie algebra.... Just do it. $\endgroup$
    – Cosmas Zachos
    Commented May 10, 2020 at 20:38

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