AI Assignment: Sphere, Matrix Operations, and QR Factorization

This assignment presents three problems related to AI and numerical methods. The first problem involves generating a sphere using vector norms. The second problem focuses on matrix operations, including equivalence, equality conditions, and the properties of matrices and their transposes. It involves creating and analyzing matrices, vectors, and functions to determine their properties. The third problem explores matrix inversion using QR factorization and Householder reflections. It requires developing functions for these methods, applying them to matrices, and comparing the results with standard matrix inversion techniques. The assignment emphasizes practical implementation, analysis, and the application of AI and mathematical concepts to solve these problems.

1. Problem 1
Objective:
To draw a sphere with R(^2).
Procedure:
To consider a vector x
P norms are declared from 1 to 100(fassumption)
Unit sphere is drawn ( [X,Y,Z] = sphere(...))
The parameters are fed into the sphere and the output values are extracted as X,Y and Z
2. Problem 2
Procedure with objectives :
Xa and Xb are vectors initialized
Matrices are declared with finite dimensions
Equivalency of the matrices should be positive(statisfied)
Equality condition of the matrices should not be positive(not statisfied)
a. The X(infinity), Xb and Kth root of X(infinity) should be dicreased respectively
b. X(infinity) and X(2) should be euqivqalent
--The X(1,2,....(infinity)) matrices are created
--The Kth root of the X(infinity) is estimated
--The region among the X(2) and X(infinity) should be developed and created
c. Two vectors are equivalent the normal in matrices are also equivalent
-The vectors are created
-The variables are estimated with a time period value say 3,6,10 values are considered
--The matrices developed and the variables with prescribed values are considered and
developed with the work accordingly
--The evaluation of the work is verified by creating a function to check the equivalency of
the work
d. Same steps of (c) is followed but the Non zero matrices have to be considered
--The non-zero matrices with equality and non-equality are to be separated
--The function have to be developed to estimate the equality of the product
e. A matrix and A transpose matrix is equivalent have to be proved
--Here we deploy a X1,X2,X3 matrix in the row
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-- The transpose of the matrix is applied
--The resultant have been analysed
--The function is developed for evaluating the equivalency in the Transpose and Input data
3. Problem 3
Objective:
Invert a Matrix using QR factorization and House holder reflections
Procedure:
Developing a function for Householder reflection
Developing a function for QR function to estimate the values
The values are estimated and calculated with the Matrix appended
Invert upper triangular function is performed
The Matrix inversion is obtained
Using 2n and 3n column the inversion of the matrix is estimated
A normal function to invert the matrix is performed and evaluated with the results obtained from
QR factorization and House holder functions
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