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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?

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