I'm (slowly) reading Fan Chung's Spectral Graph Theory. At the moment, I'm in section 1.5 which is about eigenvalues and random walks. There's a small technical point that puzzles me.
The context is a graph $G$. Let $T$ be the matrix whose $(v,v)^{\text{th}}$ entry is the degree $d_v$ of $v$. Let the volume $\operatorname{vol}(G)$ denote $\sum_v d_v$. A walk on $G$ is a finite sequence of adjacent vertices in $G$, and a random walk is given by a stochastic matrix whose $(u,v)^{\text{th}}$ entry is the probability $P(u,v)$ of walking from vertex $u$ to vertex $v$. If $f :V(G)\rightarrow \mathbb{R}$ is any initial stochastic distribution (i.e., $\sum_v f(v) =1$) then if we think of $f$ as a vector of length $|V(G)|$ the distribution after $k$ steps is simply $fP^k$. The random walk is ergodic if there is a unique stationary distribution $\pi(v)$ satisfying $\lim_{s \to \infty} fP^s(v) = \pi(v)$. This section of Chung's book is devoted to measurements of closeness between $fP^s$ and $\pi$, and estimates of how large $s$ must be for these two distributions to be close to each other.
The following is on page 17.
After $s$ steps, the relative pointwise distance (r.p.d.) of $P$ to the stationary distribution $\pi(x)$ is given by $ \Delta(s) = \max_{x,y} \ \frac{|P^s(y,x)-\pi(x)|}{\pi(x)} $ Let $\Psi_x$ denote the characteristic function of $x$ defined by: $\Psi_x(y) = \begin{cases} 1, & \text{if } y=x, \\ 0, & \text{otherwise}. \end{cases} $ Suppose $\Psi_x T^{1/2} = \sum_i \alpha_i \Phi_i $ $ \Psi_y T^{1/2} = \sum_i \beta_i \Phi_i $ where $\Phi_i$'s denote the eigenfunction of the Laplacian $\mathcal{L}$ of the weighted graph associated with the random walk. [Note: Chung discusses edge weights in this section. For the purposes of this discussion, though, I think you can just think about unweighted graphs.] In particular, $ \alpha_0 = \frac{d_x}{\sqrt{\operatorname{vol}(G)}}$ $ \beta_0 = \frac{1}{\sqrt{\operatorname{vol}(G)}}$
I cannot understand why $\alpha_0 \neq \beta_0$. It looks like $x$ and $y$ are arbitrary vertices and she's expressing the functions $\Psi_x T^{1/2}$ and $\Psi_y T^{1/2}$ that they determine in a canonical basis. Why should those representations be any different?