Master Numpy: From Array Creation to Advanced Matrix Operations

1. Numpy

In Python we commonly use the NumPy library for array creation and numerical operations. It can be installed via pip install numpy or is already available in Anaconda environments. The library is typically imported as np .

<code>import numpy as np</code>

2. Basic Data Types – array

NumPy’s fundamental data structure is the array , which is similar to Python’s list but more powerful.

2.1 Generating arrays

Arrays can be created from Python lists:

<code>>> np.array([1,2,3])
array([1, 2, 3])</code>

Multi‑dimensional arrays are also supported:

<code>>> np.array([[1,2,3],[4,5,6]])  # 2 rows, 3 columns
array([[1, 2, 3],
       [4, 5, 6]])</code>

2.2 Creating arithmetic sequences

Use linspace to generate a sequence with a specified number of elements, or arange to specify step size:

<code>>> np.linspace(1, 10, 5)
array([ 1.  ,  3.25,  5.5 ,  7.75, 10.  ])

>>> np.arange(1, 10, 2)
array([1, 3, 5, 7, 9])</code>

2.3 Shape and size

<code>>> a = np.array([[1,2,3],[4,5,6]])
>>> a.shape
(2, 3)
>>> a.size
6</code>

The array can be reshaped, e.g., to 3 rows and 2 columns:

<code>>> a.reshape((3,2))
array([[1, 2],
       [3, 4],
       [5, 6]])</code>

2.4 Indexing

<code>>> a[1,2]          # element in second row, third column
6
>>> a[:,2]          # third column
array([3, 6])
>>> a[:,[1,2]]      # second and third columns
array([[2, 3],
       [5, 6]])</code>

2.5 Filtering

<code>>> a[a > 2]
array([3, 4, 5, 6])
>>> a[(a > 2) & (a < 6)]
array([3, 4, 5])</code>

3. Universal Functions (ufuncs)

NumPy’s universal functions operate element‑wise on arrays and are more efficient than Python’s built‑in math functions.

<code>>> import math, numpy as np
>>> math.log(20)
2.995732273553991
>>> np.log(20)
2.995732273553991
>>> math.log([1,2,3])   # raises TypeError
>>> np.log([1,2,3])
array([0.        , 0.69314718, 1.09861229])</code>

4. Matrix Generation and Operations

NumPy’s linalg module provides linear‑algebra utilities.

4.1 Common matrix creation

<code>>> np.eye(3)          # identity matrix
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])
>>> np.diag([1,2,3,4]) # diagonal matrix
array([[1, 0, 0, 0],
       [0, 2, 0, 0],
       [0, 0, 3, 0],
       [0, 0, 0, 4]])
>>> np.zeros((3,4))   # 3×4 zero matrix
array([[0., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 0., 0.]])</code>

4.2 Matrix arithmetic

<code>>> A = np.array([[1,3,2],[4,1,5],[3,3,1]])
>>> B = np.array([[3,3,1],[5,2,6],[1,2,3]])
>>> A + B   # addition
array([[4, 6, 3],
       [9, 3, 11],
       [4, 5, 4]])
>>> A - B   # subtraction
array([[-2, 0, 1],
       [-1,-1,-1],
       [ 2, 1,-2]])
>>> A @ B   # matrix multiplication
array([[20, 13, 25],
       [22, 24, 25],
       [25, 17, 24]])
>>> np.linalg.inv(A)   # inverse of A
array([[-0.37837838,  0.08108108,  0.35135135],
       [ 0.2972973 , -0.13513514,  0.08108108],
       [ 0.24324324,  0.16216216, -0.2972973 ]])</code>

4.3 Determinant, eigenvalues, eigenvectors

<code>>> np.linalg.det(A)
37.0
>>> values, vectors = np.linalg.eig(A)
>>> values   # eigenvalues
array([ 7.59286438, -1.66349894, -2.92936543])
>>> vectors  # eigenvectors
array([[-0.47899104, -0.80079204,  0.50689474],
       [-0.69615171,  0.4402157 , -0.82670395],
       [-0.53473393,  0.40613082,  0.2441686 ]])</code>