Q1) Power Method and Associated problems\ i) Generate using code a random integer matrix

C

of size

4\\\\times 3

and a ma-\

trixA_(1)

defined as

A_(1)=C^(T)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.\ (0.5 marks)\ ii) Write a code in your chosen programming language to implement the\ Power method and use it to derive the largest eigenvalue

\\\\lambda _(1)

and corre-\ sponding eigenvector

x_(1)

of

A_(1)

. Find

hat(x)_(1)=(x_(1))/(||x_(1)||_(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

\\\\lambda _(1)

and

hat(x)_(1)

and a comment on the comparison.\ (0.5 marks)\ iii) Write a code to construct matrix

A_(2)=(A_(1)-hat(x)_(1)hat(x)_(1)^(T)A_(1))

. Use the Power\ method code written in ii) and use it to derive the largest eigenvalue\

\\\\lambda _(2)

and corresponding eigenvector

x_(2)

of

A_(2)

. Find

hat(x)_(2)=(x_(2))/(||x_(2)||_(2))

. Com-\ pare the values obtained in i) with these values.\ Deliverables: The first 10 iterates of

\\\\lambda _(2)

and

hat(x)_(2)

and a comment on the\ comparison.\ (0.5 marks)\ iv) Write a code to construct matrix

A_(3)=(A_(1)-hat(x)_(1)hat(x)_(1)^(T)A_(1)-(hat(x_(2)))hat(x)_(2)^(T)A_(1))

.\ Use the Power method code written in ii) and use it to derive the\ largest eigenvalue

\\\\lambda _(3)

and corresponding eigenvector

x_(3)

of

A_(3)

. Find\

hat(x)_(3)=(x_(3))/(||x_(3)||_(2))

. Compare the values obtained in i) with these values.\ Deliverables: The first 10 iterates of

\\\\lambda _(3)

and

hat(x)_(3)

and a comment on the\ comparison.\ (0.5 marks)

i) Generate using code a random integer matrix C of size 43 and a matrixA1 defined as A1=CTC and workout its characteristic equation. Using any software package, 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. (0.5 marks) 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 x^1=x12x1. 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 x^1 and a comment on the comparison. (0.5 marks) iii) Write a code to construct matrix A2=(A1x^1x^1TA1). Use the Power method code written in ii) and use it to derive the largest eigenvalue 2 and corresponding eigenvector x2 of A2. Find x^2=x22x2. Compare the values obtained in i) with these values. Deliverables: The first 10 iterates of 2 and x^2 and a comment on the comparison. (0.5 marks) iv) Write a code to construct matrix A3=(A1x^1x^1TA1x2^x^2TA1). 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=x32x3. Compare the values obtained in i) with these values. Deliverables: The first 10 iterates of 3 and x^3 and a comment on the comparison. (0.5 marks)