# MSPA 400 Session #2 Python Module #3

# Reading assignment "Think Python" either 2nd or 3rd edition:

# 2nd Edition Chapter 3 (3.4-3.9) and Chapter 10 (10.1-10.12)

# 3rd Edition Chapter 3 (pages 24-29) and Chapter 10 (pages 105-115)

# Module #3 objective: demonstrate numpy matrix calculations. For

# matrix calculations, arrays must be converted into numpy matrices.

import numpy

from numpy import *

from numpy.linalg import *

# With numpy matrices, you can add, subtract, multiply, find the transpose

# find the inverse, solve systems of linear equations and much more.

# Solve a system of consistent linear equations. Refer to Lial Section 2.5

# Example 7 Cryptography for the calculation

# Right hand side of system of equations has data entered as a list

# and converted to 3x1 matrix and then a 1x3 matrix using the transpose

# function. Similar steps are taken for the matrix A.

rhs= [96, 87, 74]

rhs=matrix(rhs)

rhs=transpose(rhs)

print ('\nRight Hand Side of Equation')

print rhs

A =[[1, 3, 4], [2, 1, 3], [4, 2, 1]]

A= matrix(A)

print ('\nMatrix A')

print A

# Numpy has various functions to perform matrix calculations. The inverse

# function inv() is one of those.

# Find inverse of A.

print ('\nInverse of A')

IA= inv(A)

print IA

# In what follows, I am converting matrices with floating point numbers to

# matrices with integer numbers. This is optional and being done to show

# that it is possible to do so with numpy matrices.

# Note that the function dot() performs matrix multiplication.

# Verify inverse by multiplying matrix A and its inverse IA.

print ('\nIdentity Matrix')

I= dot(IA,A)

I= int_(I) # This converts floating point to integer.

print I

# Solve the system of equations and convert to integer values.

# With numpy it is necessary to use dot() for the product.

result = dot(IA,rhs)

result = int_(result) # This converts floating point to integer.

print ('\nSolution to Problem')

print result

# There is a more efficient way to do this with the linalg.solve() function.

print ('\nIllustration of solution with linalg.solve(,) function')

result2= linalg.solve(A,rhs)

print int_(result2) # This converts floating point to integer.

# Some square matrices do not have inverses. The following example shows

# how this is handled with numpy. Note the magnitude of the elements.

print ('\nExample of an inverse matrix for inconsistent equations')

A= [[1,2,3],[-3,-2,-1], [-1,0,1]]

A= array(A)

IA= inv(A)

print IA

# Exercises:

# Part 1. Refer to Lial Section 2.5 Example 2. Write the code to

# reproduce the results in the example. Form the matrix A, find its inverse

# and verify such by multiplying the two to form the identity matrix.

# Show the code, matrix A, inverse of A and the Identity matrix.

# Part 2. Refer to Lial Section 2.5 page 96 problem #1. Write the code

# which solves the problem. Use linalg.solve(,).