It is known that for a true symmetric matrix $ L $ (here Graph Laplace) one can write the eigenvalue decomposition as

$$

L = U lambda U ^ { mathsf T},

$$

from where $ U $ is a *real eigenvector matrix*,

In graph signal processing papers, including the grand paper of Shuman et al. (see page 4), the adjunct (complex conjugate) of $ U $ is used to define the graph Fourier transform $ mathcal {F} _ {G} $ as

$$

has {x} = mathcal {F} _ {G} x = U ^ {*} x,

$$

from where $ x $ is the signal in vector form and $ U ^ {*} $ is the complex conjugate of $ U $,

I am curious if there is a specific reason for using the notation of *complex conjugate*?