WHY: You run two experiments recording the same neurons — but the second session used louder sounds, so ALL firing rates are 3× higher. The pattern (which neurons fire more) is identical. The intensity is different. To compare patterns, you remove the magnitude.

That's normalization: divide every component by the vector's length. The result is a unit vector — same direction, length = exactly 1.

Key insight from NMA: Normalizing does NOT change direction. The arrow still points the same way — it just gets shrunk to length 1.

Before and after normalization: same direction, length becomes 1.

x̂ = x / ‖x‖ Divide each component by the length → unit vector with length 1.

x̂ is pronounced 'x-hat'. The hat symbol means 'unit vector'.

import numpy as np

v = np.array([3, 4])           # length = 5
v_unit = v / np.linalg.norm(v) # normalize

print(v_unit)                  # [0.6, 0.8]
print(np.linalg.norm(v_unit))  # 1.0 — always 1 after normalization

Session A neurons: [10, 50, 2] (quiet lab)
Session B neurons: [30, 150, 6] (loud lab — 3× everything)

Both normalize to the same unit vector → you can now compare the pattern of activity, not the volume.