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Topic: Python SciPy – From Easy to Top: Part 4 of 6: Linear Algebra with SciPy

---

1. Introduction to Linear Algebra in SciPy

• Linear algebra is fundamental in scientific computing, machine learning, and data science.

• SciPy provides advanced linear algebra routines built on top of LAPACK and BLAS libraries.

• The main sub-package is scipy.linalg which extends NumPy’s linear algebra capabilities.

---

2. Basic Matrix Operations

You can create matrices using NumPy arrays:

import numpy as np

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

---

3. Matrix Addition and Multiplication

# Addition
C = A + B
print("Matrix Addition:\n", C)

# Element-wise Multiplication
D = A * B
print("Element-wise Multiplication:\n", D)

# Matrix Multiplication
E = np.dot(A, B)
print("Matrix Multiplication:\n", E)

---

4. Using `scipy.linalg` for Advanced Operations

Import SciPy linear algebra module:

from scipy import linalg

---

5. Matrix Inverse

Calculate the inverse of a matrix (if invertible):

inv_A = linalg.inv(A)
print("Inverse of A:\n", inv_A)

---

6. Determinant

Calculate the determinant:

det_A = linalg.det(A)
print("Determinant of A:", det_A)

---

7. Eigenvalues and Eigenvectors

Find eigenvalues and eigenvectors:

eigvals, eigvecs = linalg.eig(A)
print("Eigenvalues:\n", eigvals)
print("Eigenvectors:\n", eigvecs)

---

8. Solving Linear Systems

Solve Ax = b where b is a vector:

b = np.array([5, 11])
x = linalg.solve(A, b)
print("Solution x:\n", x)

---

9. Singular Value Decomposition (SVD)

Decompose matrix A into U, Σ, and V^T:

U, s, VT = linalg.svd(A)
print("U matrix:\n", U)
print("Singular values:", s)
print("V^T matrix:\n", VT)

---

10. LU Decomposition

Decompose matrix A into lower and upper triangular matrices:

P, L, U = linalg.lu(A)
print("P matrix:\n", P)
print("L matrix:\n", L)
print("U matrix:\n", U)

---

11. QR Decomposition

Factorize A into Q and R matrices:

Q, R = linalg.qr(A)
print("Q matrix:\n", Q)
print("R matrix:\n", R)

---

12. Norms of Vectors and Matrices

Calculate different norms:

# Vector norm
v = np.array([1, -2, 3])
norm_v = linalg.norm(v)
print("Vector norm:", norm_v)

# Matrix norm (Frobenius norm)
norm_A = linalg.norm(A, 'fro')
print("Matrix Frobenius norm:", norm_A)

---

13. Checking if a Matrix is Positive Definite

Try Cholesky decomposition:

try:
    L = linalg.cholesky(A)
    print("Matrix is positive definite")
except linalg.LinAlgError:
    print("Matrix is not positive definite")

---

14. Summary

• SciPy’s linalg module provides extensive linear algebra tools beyond NumPy.

• Operations include inverse, determinant, eigenvalues, decompositions, and solving linear systems.

• These tools are essential for many scientific and engineering problems.

---

Exercise

• Compute the eigenvalues and eigenvectors of the matrix \[\[4, 2], \[1, 3]].

• Solve the system of equations represented by:

  2x + 3y = 8

  5x + 4y = 13

• Perform SVD on the matrix \[\[1, 0], \[0, -1]] and explain the singular values.

---

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Python tip:
Use np.polyval() to evaluate a polynomial at specific values.

import numpy as np
poly_coeffs = np.array([3, 0, 1]) # Represents 3x^2 + 0x + 1
x_values = np.array([0, 1, 2])
y_values = np.polyval(poly_coeffs, x_values)
print(y_values) # Output: [ 1  4 13] (3*0^2+1, 3*1^2+1, 3*2^2+1)

Python tip:
Use np.polyfit() to find the coefficients of a polynomial that best fits a set of data points.

import numpy as np
x = np.array([0, 1, 2, 3])
y = np.array([0, 0.8, 0.9, 0.1])
coefficients = np.polyfit(x, y, 2) # Fit a 2nd degree polynomial
print(coefficients)

Python tip:
Use np.clip() to limit values in an array to a specified range, as an instance method.

import numpy as np
arr = np.array([1, 10, 3, 15, 6])
clipped_arr = arr.clip(min=3, max=10)
print(clipped_arr)

Python tip:
Use np.squeeze() to remove single-dimensional entries from the shape of an array.

import numpy as np
arr = np.zeros((1, 3, 1, 4))
squeezed_arr = np.squeeze(arr) # Removes axes of length 1
print(squeezed_arr.shape) # Output: (3, 4)

Python tip:
Create a new array with an inserted axis using np.expand_dims().

import numpy as np
arr = np.array([1, 2, 3]) # Shape (3,)
expanded_arr = np.expand_dims(arr, axis=0) # Add a new axis at position 0
print(expanded_arr.shape) # Output: (1, 3)

Python tip:
Use np.ptp() (peak-to-peak) to find the range (max - min) of an array.

import numpy as np
arr = np.array([1, 5, 2, 8, 3])
peak_to_peak = np.ptp(arr)
print(peak_to_peak) # Output: 7 (8 - 1)

Python tip:
Use np.prod() to calculate the product of array elements.

import numpy as np
arr = np.array([1, 2, 3, 4])
product = np.prod(arr)
print(product) # Output: 24 (1 * 2 * 3 * 4)

Python tip:
Use np.allclose() to compare two arrays for equality within a tolerance.

import numpy as np
a = np.array([1.0, 2.0])
b = np.array([1.00000000001, 2.0])
print(np.allclose(a, b)) # Output: True

Python tip:
Use np.array_split() to split an array into N approximately equal sub-arrays.

import numpy as np
arr = np.arange(7)
split_arr = np.array_split(arr, 3) # Split into 3 parts
print(split_arr)

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