-\Re{\rm e}\mathcal{F}\Im{\rm m}\mathcal{F}^{-1} & \Im{\rm m}\mathcal{F}^{-1}  \\
\end{array}
\right)\, .$$

A direct computation of $|\mathcal{Z}_{\rm fix}|^{2}$ gives

$$|\mathcal{Z}_{\rm fix}|^{2} = 
|\langle\, \mathcal{V}_{\rm fix} \mid q \, \rangle|^{2}
=-\langle\, q \mid \mathcal{V}_{\rm fix}\, \rangle
\langle\, \mathcal{V}^{*}_{\rm fix}\mid q\, \rangle\, .$$

The matrix of this bilinear is

$$\mid \mathcal{V}_{\rm fix}\, \rangle
\langle\, \mathcal{V}^{*}_{\rm fix}\mid =
-\left( 
  \begin{array}{cc}
    \mathcal{M}_{\Lambda}\mathcal{M}^{*}_{\Sigma} &  
-\mathcal{M}_{\Lambda}\mathcal{L}^{*\, \Sigma} \\
& \\
    -\mathcal{L}^{\Lambda}\mathcal{M}^{*}_{\Sigma} &  
\mathcal{L}^{\Lambda}\mathcal{L}^{*\, \Sigma} \\
  \end{array}
\right)_{\rm fix}\, .$$

We can use the relation

$$\mathcal{L}^{*\, \Lambda}\mathcal{L}^{\Sigma}
=
-\textstyle{1\over 2}\Im{\rm m}(\mathcal{N})^{-1|\Lambda\Sigma}
-f^{\Lambda}{}_{i}\mathcal{G}^{ii^{*}}
f^{*\, \Sigma}{}_{i^{*}}\, ,$$

taking into account that at the fixed point the second term in the r.h.s. will not contribute, and that only its symmetric part will contribute, to get [@Ceresole:1995ca; @Ferrara:1996dd]

$$|\mathcal{Z}_{\rm fix}|^{2}
=-{\textstyle\frac{1}{2}}
q^{T}\mathcal{M}(\mathcal{N}_{\rm fix})q\, .$$

So far we have checked that the coefficient of the $\rho^{2}$ term of $-g_{rr}$ is given by the value of the central charge at the fixed point but, if there are terms of higher order in $\rho$ in $-g_{rr}$ there will not be a regular horizon. We can, however, see that taking another derivative of $-g_{rr}$ w.r.t. $\rho$ at $\rho=0$ will give zero if the attractor equations (\[eq:attractorequations\]) are satisfied and the same will happen for higher derivatives.

Summarizing 