SciPy Linear Algebra

SciPy is built upon the ATLAS LAPACK and BLAS library, and it provides very fast linear algebra capabilities. Linear algebra routine accepts two-dimension array object and output is also given as a two-dimension array. If we want more speed in computation, then we have to dig deep in this scenario.

A linear algebra problem can be solved by typing the following scipy function:

Linear Equation

The linalg.solve is used to solve the linear equation a*x + b*y = Z, for the unknown x, y values.

x + 3y +10z = 10
2x + 12y + 7z = 18
5x + 8y + 8z = 30

Here we will solve above linear equation by using the linear.solve command for the faster calculation.

import numpy as np
from scipy import linalg
# We are trying to solve a linear algebra system which can be given as 
#         x + 3y +10z = 10
#         2x + 12y + 7z = 18
#         5x + 8y + 8z = 30
# Creating input array
a = np.array([[1, 3, 10], [2, 12, 7], [5, 8, 8]])
# Solution Array
b = np.array([[10], [18], [30]])
# Solve the linear algebra
x = linalg.solve(a, b)
# Print results
print(x)
# Checking Results
print("\n Checking results, Vectors must be zeros")
print(a.dot(X) - b)

Output:

[[4.55393586]
 [0.51311953]
 [0.39067055]]

 Checking results, Vectors must be zeros 
[[0.]
 [0.]
 [0.]]

In the above program, we have declared a and b as variable where a stored coefficients of equation and b stored the right-hand-side value. Variable x stored the evaluated solution.

Finding the determinants

The determinant of the square matrix is found by using the linalg.det() function. The determinate A is often denoted as |A| in the linear algebra. It accepts a matrix and returns a scalar value.

Let's consider the following example:

from scipy import linalg
import numpy as np
#Declaring the numpy array
A = np.array([[5,9],[8,4]])
#Passing the values to the det function
x = linalg.det(A)
#printing the result
print(x)

Output:

-52

Eigenvalues and Eigenvectors

Finding eigenvalues and eigenvector problems are the most common problem in linear algebra. We can find the Eigenvalues (?) and the corresponding Eigenvectors (v) of a square matrix (A) by linalg.eig() function. Consider the following example:

Av = λv

#importing the scipy and numpy packages
from scipy import linalg
import numpy as np
#Declaring the numpy array
a = np.array([[3,2],[4,6]])
#Passing the values to the eig function
l, v = linalg.eig(a)
#printing the result for eigenvalues
print(l)
#printing the result for eigenvectors
print(v)

Output:

[-0.37228132+0.j  5.37228132+0.j]
[[-0.82456484 -0.41597356]
 [ 0.56576746 -0.90937671]]

SciPy svd

The svd is stands for single value decomposition. The unique value decomposition of a matrix A is the factorization of A into the product of three matrices A = UDVT, where the columns of U and V are orthonormal, and the matrix D is diagonal with real positive entries.

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