The equation

$$\mathbf{A x}=\lambda \mathbf{x}$$

where $\mathbf{A}$ is a real square matrix and $\mathbf{x}$ a column vector, has a simple eigenvalue $\lambda=\mu$ with corresponding right-eigenvector $\mathbf{x}=\mathbf{X}$. Show how to find expressions for the perturbed eigenvalue and right-eigenvector solutions of

$$\mathbf{A} \mathbf{x}+\epsilon \mathbf{b}(\mathbf{x})=\lambda \mathbf{x}, \quad|\epsilon| \ll 1$$

to first order in $\epsilon$, where $\mathbf{b}$ is a vector function of $\mathbf{x}$. State clearly any assumptions you make.

If $\mathbf{A}$ is $(n \times n)$ and has a complete set of right-eigenvectors $\mathbf{X}^{(j)}, j=1,2, \ldots n$, which span $\mathbb{R}^{n}$ and correspond to separate eigenvalues $\mu^{(j)}, j=1,2, \ldots n$, find an expression for the first-order perturbation to $\mathbf{X}^{(1)}$ in terms of the $\left\{\mathbf{X}^{(j)}\right\}$ and the corresponding lefteigenvectors of $\mathbf{A}$.

Find the normalised eigenfunctions and eigenvalues of the equation

$$\frac{d^{2} y}{d x^{2}}+\lambda y=0,0<x<1$$

with $y(0)=y(1)=0$. Let these be the zeroth order approximations to the eigenfunctions of

$$\frac{d^{2} y}{d x^{2}}+\lambda y+\epsilon b(y)=0,0<x<1$$

with $y(0)=y(1)=0$ and where $b$ is a function of $y$. Show that the first-order perturbations of the eigenvalues are given by

$$\lambda_{n}^{(1)}=-\epsilon \sqrt{2} \int_{0}^{1} \sin (n \pi x) \quad b(\sqrt{2} \sin n \pi x) d x$$

Part II