Graph Fourier transform: the adjoint notation for the eigenbasis matrix

Question

I already asked this question here but there is no response. I'd like to ask this question in signal processing domain.

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?

Answer 1

You terminology is something I am not used to.

$$ \hat{x} = \mathcal{F}_{G} x $$

If by $\hat{x}$ you mean the DFT (complex except very special cases), $x$ is your input (real or complex), and $\mathcal{F}_{G}$ is the matrix composed of the sinusoidal basis vectors, then your assertion that $U$ is real is dubious. $\mathcal{F}_{G}$ is most certainly not. With the proper normalizattion:

$$ \mathcal{F}_{G}^{-1} = \mathcal{F}_{G}^{*} $$

Now, for a known $x$ where $\hat{x}$ is known to be complex, how can $U$ be real?

Since they are using the complex conjugate, I would say that U is expected to also be capable of being complex. I have not read the paper so I don't know where you picked up that assumption.