Centering Matrix

Given a vector $v$, with mean value of its elements $m$, we can center the vector by subtracting the mean $m$ from each element,

import numpy as np

n = 10

v = np.random.randn(n)
v_c = v - v.mean()

This operation is easy and obvious. However, the formalism is not elegant. In some cases, we would like to formulate the process of centering the elements as operators,

$$ v_c = \operatorname{\hat H}v. $$

In this case, the operator $\operatorname{\hat H}$ is simply a matrix

$$ \operatorname{\hat H} \to I_n - \frac{1}{n} J_n, $$

where $n$ is the dimension of the vector $v$, $I_n$ is a identity matrix, $J_n$ is a matrix of all $1$s.

cm = np.identity(n) - np.ones((n, n)) / n
np.matmul(cm, v)