Vector Norm (np.linalg.norm)

How "big" is a player's stat vector? How "far apart" are Virat and Rohit statistically? Both questions are answered by a norm — the length of a vector. Norms appear everywhere in machine learning: distances in k-nearest-neighbours, normalising features, regularisation (L1/L2) and cosine similarity.

Definition

A norm is a function that assigns a non-negative length to a vector. The default Euclidean (L2) norm of v is ‖v‖ = √(v[0]² + v[1]² + … + v[n-1]²) — the straight-line distance from the origin to the point v, exactly like Pythagoras' theorem.

• L2 norm (ord=2, default for vectors) — square, sum, square root. "As the crow flies".
• L1 norm (ord=1) — sum of absolute values. "Manhattan / taxi-cab distance" along grid streets.
• Max (∞) norm (ord=np.inf) — the largest absolute value.
• Distance between two vectors = norm of their difference: np.linalg.norm(a - b).
• Unit vector — a vector with norm 1, made by dividing a vector by its norm (normalising).

Syntax

np.linalg.norm(x, ord=None, axis=None, keepdims=False)

Example — L2, L1 and max norm of (3, 4)

import numpy as np
v = np.array([3, 4])
print(np.linalg.norm(v))
print(np.sqrt(np.sum(v ** 2)))
print(np.linalg.norm(v, ord=1))
print(np.linalg.norm(v, ord=np.inf))

5.0
5.0
7.0
4.0

How it works:

      y
    4 ┤        • (3, 4)
      │      ╱ │
      │ L2=5╱  │ 4        L2  = √(3² + 4²) = √25 = 5
      │   ╱    │          L1  = |3| + |4|       = 7
      │ ╱      │          max = max(|3|, |4|)   = 4
    0 ┼────────┴──── x
      0        3

The second line proves the definition: norm(v) is just sqrt(sum(v**2)), which also equals np.sqrt(v @ v).

Example — distance between two players and normalising

virat = np.array([45.0, 140.0])   # [batting average, strike rate]
rohit = np.array([30.0, 145.0])
print(np.linalg.norm(virat - rohit).round(3))
unit = virat / np.linalg.norm(virat)
print(unit.round(4), np.linalg.norm(unit))

15.811
[0.306 0.952] 1.0

The Euclidean distance is √(15² + 5²) = √250 ≈ 15.811. Dividing by the norm gives a unit vector (length exactly 1) pointing in the same direction — used before cosine similarity and in many ML pipelines.

Tip: when features have very different scales (average ≈ 40, strike rate ≈ 140), the larger one dominates the distance. Standardise features first in real projects.

Example — matrix norm and row-wise norms

M = np.array([[1, 2], [3, 4]])
print(np.linalg.norm(M).round(4))
print(np.linalg.norm(M, axis=1).round(4))
print(np.linalg.norm(M, axis=0).round(4))

5.4772
[2.2361 5.    ]
[3.1623 4.4721]

• No axis → the Frobenius norm: √(1 + 4 + 9 + 16) = √30 ≈ 5.4772 — all entries treated as one long vector.
• axis=1 → one L2 norm per row (each player's vector length): √5, √25.
• axis=0 → one per column.

Comparison — the common norms

Common mistakes

• ❌ Writing np.sqrt(v ** 2) and forgetting the sum (gives element-wise absolute values). ✅ np.sqrt(np.sum(v ** 2)) or simply np.linalg.norm(v).
• ❌ np.linalg.norm(a) - np.linalg.norm(b) as a "distance". ✅ Distance is np.linalg.norm(a - b).
• ❌ Forgetting axis=1 when you want one length per row — you get a single number for the whole matrix. ✅ Pass axis.
• ❌ Normalising a zero vector (division by 0 → nan). ✅ Check the norm is non-zero first.

Interview tip

Q: What is the difference between the L1 and L2 norm? A: L2 is the square root of the sum of squares (straight-line length) and penalises large values heavily; L1 is the sum of absolute values (grid distance) and is less sensitive to outliers. In regularisation, L1 (lasso) pushes weights to exactly zero, while L2 (ridge) shrinks them smoothly.