Instructions

• Use python programming language.
• By random entries, I mean a system generated random number.

Matrix decompositions

a) Consider a symmetric and positive definite matrix A of size n n. Write a code to construct an elementary matrix for every elementary row operation that is performed on A and using the theory explained in the class write the decomposition of A as LU , where L is a lower triangular matrix and U is an upper triangular matrix.

Deliverables: The code snippet showing the generation of elementary matrices for a given elementary row operation and getting L and U , and verification of A = LU.

b) For the above problem, work out the Choleskys decomposition.

Deliverables: The code snippet showing the generation of L and verification of A = LLT.

c) Given n linearly independent vectors in m dimensions (the corresponding matrix with these as columns is A of size m n), with m > n, read about the QR decomposition of a matrix and generate Q and R.

Deliverables: The code snippet generating Q and R.

d) Take a random 5 x 4 matrix having all its columns as linearly independent and it's decompose into Q and R. What is your observation on the diagonal elements of R?

Deliverables: The random matrix, Q and R and your observation on the diagonal elements of R