# Graph Fourier transform: the adjoint notation for the eigenbasis matrix

It is well-known that for a real symmetric matrix $$L$$ (here, graph Laplacian) one can write the eigenvalue decomposition as

$$L = U \Lambda U^{\mathsf T},$$ where $$U$$ is a real eigenvector matrix. Moreover, in graph signal processing papers, including the great paper by Shuman et al. (cf. page 4), the adjoint (complex conjugate) of $$U$$ is used to define the graph Fourier transform $$\mathcal{F}_{G}$$ as $$\hat{x} = \mathcal{F}_{G} x = U^{*}x,$$ where $$x$$ is the signal in vector form and $$U^{*}$$ is the complex conjugate of $$U$$.

I am curious to know is there any specific reason for using the notation of complex conjugate since $$U$$ is real?