Correlation matrix, and why correlation is not causation

The problem, from first principles

03-probability/03-covariance-correlation computed correlation between exactly two variables. A real dataset has dozens or hundreds of features, and the practical question is never just "are A and B related," it's "which PAIRS, among everything I have, are related, and how strongly." Checking every pair by hand doesn't scale, you need the full grid at once: every feature's correlation with every other feature, in one table.

That table, the correlation matrix, is one of the very first things a practitioner looks at when exploring a new dataset: it surfaces redundant features (two columns measuring almost the same thing), hints at which features might matter for a prediction target, and, done carelessly, tempts a very common and genuinely dangerous misinterpretation this question's Theory section exists specifically to head off.

From theory to code

Theory generalizes 03-probability/03-covariance-correlation's single-pair formula to every pair at once: a symmetric matrix with 1.0 on the diagonal (each feature trivially correlates perfectly with itself) and the pairwise correlation everywhere else. Implement the full matrix by looping over every unique pair once (exploiting symmetry), then a helper that finds the single strongest off-diagonal relationship.

Implement correlation_matrix(x) and most_correlated_pair(corr_matrix) against that reasoning. The signatures and docstrings are already in the editor.

Constraints

• x is (num_samples, num_features); the result is (num_features, num_features).
• The diagonal is always exactly 1.0.
• The matrix must be symmetric: result[i, j] == result[j, i].
• most_correlated_pair excludes the diagonal and compares by absolute value (a strong negative correlation counts as "strongly related" too).

Hints

Open one at a time. Each gives away a little more than the last.