Explain the concepts with example with detailed step by step explanation.

Python code without any inbuilt library.

i$)$ Generate using code a random integer matrix C of size $43$ and a matrix A$1$ defined as A$1$

$=CTC$ and workout its characteristic equation. Determine the eigenvalues and

eigenvectors.

Deliverables: The matrices $C$ and $A1,$ the computation of the characteristic 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 $1$ and corresponding eigenvector $x1$ of $A1.$

Find ${?}^{n}x1=x1||x1||2_{{?}_{?}}.$ 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 $1$ and ${?}^{{?}^{{?}^{?}}}x1$ and $a$

comment on the comparison.

iii$)$ Write a code to construct matrix A$2=(A1-x^{{?}^{{?}^{?}}}1x^{{?}^{{?}^{?}}}T1A1).$ Use the Power method code

written in ii$)$ and use it to derive the largest eigenvalue $2$ and corresponding eigenvector

$x2$ of A$2.$ Find $^x2=x2||x2||.$ Comqpare the values obtained in i$)$ with these values.

Deliverables: The first $10$ iterates of $2$ and ${?}^{{?}^{{?}^{?}}}x2$ and a comment on the comparison.

iv$)$ iv$)$ Write a code to construct matrix $A3=(A1-x^{{?}^{{?}^{?}}}1x^{{?}^{{?}^{?}}}T1A1-x^{{?}^{{?}^{?}}}2x^{{?}^{{?}^{?}}}T2A1).$ Use the

Power method code written in ii$)$ and use it to derive the largest eigenvalue $3$ and

corresponding eigenvector $x3$ of $A3.$ Find $x^{{?}^{{?}^{?}}}3=x3||x3||2_{{?}_{?}}.$ Compare the values obtained

in i$)$ with these values.

Deliverables: The first $10$ iterates of $3$ and ${?}^{{?}^{{?}^{?}}}x3$ and a comment on the comparison.