Eigenvector expansion?

Question

Does anyone know what they mean by Eigenvector expansion of $u_0$? I am at a complete loss on that part of the problem. The coding part I understood :)

Answer 1

Assuming you're considering a finite-difference based numerical method, with a uniform grid $0=x_0<x_1<\cdots<x_{N-1}<x_n=1$, then $u^0 = [u(x_1,0), ~ u(x_2,0), ~ \ldots, ~ u(x_{N-1},0)]^\top \in \mathbb{R}^{N-1}$.

Let $C = I - \beta k D_+ D_-$. (Then $u^{n+1} = C u^n$, or equivalently $u^n = C^n u^0$.)

Suppose $C$ is diagonalizable, with eigenpairs $(\lambda_j, \boldsymbol\xi_j)$ ($j=1,2,\ldots,N-1$). The hint -- which asks you to "expand $u^0$ in terms of the eigenvectors of $C$" -- is asking you to compute the coefficients $c_1, c_2, \ldots, c_{N-1}$ satisfying $$u^0 = c_1 \boldsymbol\xi_1 + c_2 \boldsymbol\xi_2 + \cdots + c_{N-1} \boldsymbol\xi_{N-1}.$$