# Eigenvalues of linearized RG

Consider RG transformation in the vicinity of a fixed point: $$K_n^\prime=K_n^* + \delta K_{n}^\prime = K_n^* + \sum_m \frac{\partial K^\prime_n}{\partial K_m} \delta K_m + O(\delta K^2)$$ where $$M_{nm} = \frac{\partial K^\prime_n}{\partial K_m}$$ is the matrix of linearized RG.

Using semigroup property of RG: $$R_{l_1} R_{l_2} = R_{l_2} R_{l_1} = R_{l_1 l_2}$$ where $$l$$ is the scale we choose to integrate out.

we have $$M^{l_1} M^{l_2} = M^{l_2} M^{l_1} = M^{l_1 l_2} \\ \Lambda^{(\sigma)}_{l_1} \Lambda^{(\sigma)}_{l_2} = \Lambda^{(\sigma)}_{l_1 l_2}$$ where $$\Lambda_l^{(\sigma)}$$ is the eigenvalues of $$M^l$$, $$\sigma$$ used to label different eigenvalues.

Using above equation of eigenvalues, it can be shown that $$\Lambda_l^{(\sigma)} = l^{y_{\sigma}}$$

Here is my question, from above form it seems that $$\Lambda_l$$ cannot be negative, but in some problem, the eigenvalues are negative through concrete calculation. How can it be possible?