Compute Covariance Matrix

Name: TensorTonic
Author: TensorTonic

Linear AlgebraData Processing

Easy

Compute the sample covariance matrix without using np.cov. First center each feature:

X_c = X - \mu

Then compute:

\Sigma = \frac{X_c^{\mathsf T}X_c}{N-1}

Here, $X$ has $N$ samples and $D$ features, $\mu$ is the vector of feature means, $X_c$ is the centered data, and $\Sigma$ is the $D \times D$ sample covariance matrix. Return $\Sigma$ as a NumPy array.

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Examples

Input: X = [[1, 2], [2, 3], [3, 4]]

Output: [[1.0, 1.0], [1.0, 1.0]]

Explanation: Both features vary together by the same amount after centering.

Input: X = [[1, 0], [0, 1]]

Output: [[0.5, -0.5], [-0.5, 0.5]]

Hint 1

Use X - np.mean(X, axis=0) to center every feature.

Hint 2

Use centered.T @ centered before dividing by X.shape[0] - 1.

Requirements

Center each feature independently
Divide by N minus 1 for sample covariance
Do not use np.cov
Return a NumPy array of shape $(D,D)$

Constraints

X is a rectangular numeric list with shape $(N,D)$
$2 \le N \le 10{,}000$
Use NumPy only

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