# Eigenvalues and more of $\lambda^2$, where $\lambda$ is a general $su(3)$ matrix.

Consider a general $$su(3)$$ matrix in usual Gell-Mann basis $$\lambda = l_a \lambda_a$$, with $$a=1,...,8$$. Are there any resources studying the eigenvalues /eigenvectors of $$\lambda^2 = \frac{2}{3} l^2 I + d^{abc}l_a l_b \lambda_c \quad ?$$ For one, I have computed the eigenvalues (which should be real, given $$\lambda$$ is hermitian) with the help of Mathematica and certain $$su(3)$$ structure constant identities to be $$\lambda_1 = \frac{2}{3} l^2 - \frac{1}{3}R^{1/3}- \frac{1}{3}l^4R^{-1/3}\\ \lambda_2 = \frac{2}{3} l^2 + \frac{1}{6}(1-i \sqrt{3})R^{1/3}+ \frac{1}{6\sqrt{3}}(3 i + \sqrt{3})l^4R^{-1/3}\\ \lambda_3 = \frac{2}{3} l^2 + \frac{1}{6}(1+i \sqrt{3})R^{1/3}+ \frac{1}{6\sqrt{3}}(-3 i + \sqrt{3})l^4 R^{-1/3},$$ where $$R=-6(d^{abc}l_al_bl_c)^2 +l^6+2|d^{abc}l_al_bl_c|\sqrt{3(d^{abc}l_al_bl_c)^2-l^6}$$ But I'm not entirely sure I can trust these or if there are simpler, less cryptic ways of writing them. I have learned from eq. 3.14 from doi:10.1007/BF01654302 that the argument of the square root is always negative but Mathematica complains about taking cubed roots when I try to give it numerical values respecting that (which I imagine comes from secret assumptions it made while solving the cubic characteristic equation).

finding eigenvectors seems rather difficult..

• Interesting. But what is the question? – lcv May 23 at 1:24
• Principally, I'm looking for resources on this material. If anyone has diagonalised $\lambda^2$ in the literature... – Rudyard May 23 at 8:54
• I would state it more clearly in the text what you're looking for. I think you would get more response. $\lambda$ is the most general traceless $3\times3$ matrix. It's eigevalues, in general, will be roots of a cubic (summing to zero). The same holds for $\lambda^2$, except that it's not traceless. You seek an expression in terms of $\ell$s? You can easily do a numerical check yourself. – lcv May 23 at 18:09