In this chapter, we introduced the matrix \(\mathbb{D}\) defined as the derivative acting on a grid with spacing \(\Delta x\). This matrix has the form

\[\mathbb{D}=\left(\begin{array}{ccccc}\ddots & \vdots & \vdots & \vdots & \cdots \tag{2.87}\\\cdots & 0 & \frac{1}{2 \Delta x} & 0 & \cdots \\\cdots & -\frac{1}{2 \Delta x} & 0 & \frac{1}{2 \Delta x} & \cdots \\\cdots & 0 & -\frac{1}{2 \Delta x} & 0 & \cdots \\\vdots & \vdots & \vdots & \vdots & \ddots\end{array}\right)\]

with only non-zero entries immediately above and below the diagonal. In this problem, we will just study the \(2 \times 2\) and \(3 \times 3\) discrete derivative matrices.

(a) Explicitly construct \(2 \times 2\) and \(3 \times 3\) discrete derivative matrices, according to the convention above.

(b) Now, calculate the eigenvalues of both of these matrices. Note that there should be two eigenvalues for the \(2 \times 2\) matrix and three eigenvalues for the \(3 \times 3\) matrix. Are any of the eigenvalues 0 ? For those that are non-zero, are the eigenvalues real, imaginary, or general complex numbers?

(c) Determine the eigenvectors of this discrete derivative matrix, for each eigenvalue. Make sure to normalize the eigenvectors. Are eigenvectors corresponding to distinct eigenvalues orthogonal?

(d) Now, consider exponentiating the discrete derivative matrix to move a distance \(\Delta x\). Call the resulting matrix \(\mathbb{M}\) :

\[\begin{equation*}\mathbb{M}=e^{\Delta x \mathbb{D}} \tag{2.88}\end{equation*}\]

What is the result of acting this exponentiated matrix on each of the eigenvectors that you found in part (c)?

Consider the Taylor expansion of this exponential. How does \(\mathbb{D}^{n}\) act on an eigenvector of \(\mathbb{D}\) ?

(e) Now, just evaluate the exponentiated derivative matrices that you are studying in this problem. That is, determine the closed form of the \(2 \times 2\) and \(3 \times 3\) matrices \(\mathbb{M}\), where \[\begin{equation*}\mathbb{M}=e^{\Delta x \mathbb{D}} \tag{2.89}\end{equation*}\]

Can you write down a recursive formula for \(\mathbb{D}^{n}\) that appears in the Taylor expansion of the exponential?