The power method and its variants

Find a dominant eigenpair for $\mathbf{A} = \begin{pmatrix} 7 & 2 & 3 \\ 0 & 2 & 0 \\ -6 & -2 & -2 \end{pmatrix}$ by the power method.

Apply the inverse power method to find an eigenvector for each of the eigenvalues of the matrix $\mathbf{A}$ in Exercise 7.3.13.

Explain why the function $m(\overrightarrow {v})$ used in the development of the power method is not a continuous function, so statements like $m(\overrightarrow {x}_n) \to m(\overrightarrow {x})$ when $\overrightarrow {x}_n \to \overrightarrow {x}$ are not valid. Nevertheless, if $\lim_{n\to \infty } \overrightarrow {x}_n \ne \overrightarrow {0}$, then $\lim_{n\to \infty } m(\overrightarrow {x}_n) \ne 0$.

Let $A$ be an $n \times n$ normal matrix over $\mathbb {C}$. How would one apply genetic algorithms to find the eigenvalues of $A$. This means we have to construct a fitness function $f$ with the minima as the eigenvalues. The eigenvalue equation is given by $A\overrightarrow {x}=z\overrightarrow {x}$ ($z \in \mathbb {C}$ and $\overrightarrow {x} \in \mathbb {C}^{n}$ with $\overrightarrow {x} \neq \overrightarrow {0}$). The characteristic equation is $p(z) \equiv \operatorname {det}\left(A-zI_{n}\right)=0$. What would be a fitness function? Apply it to the matrices

The power method is the simplest algorithm for computing eigenvectors and eigenvalues. Consider the vector space $\mathbb {R}^{n}$ with the Euclidean norm $\left\| \overrightarrow {x}\right\|$ of a vector $\overrightarrow {x} \in \mathbb {R}^{n}$. The iteration is as follows: Given a nonsingular $n \times n$ matrix $M$ and a vector $\overrightarrow {x}_{0}$ with $\left\| \overrightarrow {x}_{0}\right\| =1$. One defines

This defines a dynamical system on the sphere $S^{n-1}$. Since $M$ is invertible we have