Generate using code a random integer matrix C of size $4\backslash$times $3$ and a ma$-$

trix A$1$ defined as A$1=$ CT C and workout its characteristic equation.

Using any software package, determine the eigenvalues and eigenvec$-$

tors.

Deliverables: The matrices C and A$1,$ the computation of the char$-$

acteristic equation, the eigenvalues and eigenvectors as obtained from

the package.

ii$)$ Write a code in your chosen programming language to implement the

Power method and use it to derive the largest eigenvalue $\backslash$lambda $1$ and corre$-$

sponding eigenvector x$1$ of A$1.$ Find $$x$1=$ x$1/\backslash |$ x$1\backslash |2$

$.$ Compare the values obtained in i$)$ with these values.

Deliverables: The handwritten code that implements the Power method,

the first $10$ iterates of eigenvalue generated by the algorithm and the

final $\backslash$lambda $1$ and $$x$1$ and a comment on the comparison.

iii$)$ Write a code to construct matrix A$2=($A$1$x$1$x$1$TA$1).$ Use the Power

method code written in ii$)$ and use it to derive the largest eigenvalue

$\backslash$lambda $2$ and corresponding eigenvector x$2$ of A$2.$ Find $$x$2=$ x$2/\backslash |$ x$2\backslash |2$

$.$ Com$-$pare the values obtained in i$)$ with these values.

Deliverables: The first $10$ iterates of $\backslash$lambda $2$ and $$x$2$ and a comment on the

comparison.

iv$)$ Write a code to construct matrix A$3=($A$1$x$1$x$1$TA$1$x$2$xT$2$A$1).$

Use the Power method code written in ii$)$ and use it to derive the

largest eigenvalue $\backslash$lambda $3$ and corresponding eigenvector x$3$ of A$3.$ Find

$$x$3=$ x$3/\backslash |$ x$3\backslash |2$

$.$ Compare the values obtained in i$)$ with these values.

Deliverables: The first $10$ iterates of $\backslash$lambda $3$ and $$x$3$ and a comment on the

comparison.