Trace of a Matrix
The problem, from first principles
The trace is the smallest, cheapest-to-compute summary a square matrix has — just its diagonal entries, added up. It sounds almost too simple to matter, but it shows up constantly: as a fast way to compute the sum of a matrix's eigenvalues without finding any of them, inside the KL divergence formula for two multivariate Gaussians, and inside regularization terms that penalize a weight matrix's overall scale.
From theory to code
Implement trace(A), summing A's diagonal entries, then trace_of_product(A, B), computing trace(A @ B) — but doing it in a way that reveals a useful shortcut identity, covered in Theory. The signatures and docstrings are already in the editor.
Constraints
Ais a square 2D NumPy array (n × n).- For
trace_of_product,Ais(m, n)andBis(n, m), soA @ Bis a valid(m, m)square matrix.
Hints
Hint 1
trace(A) is Σ_{i} A[i, i] — 01-summation-notation's Σ applied directly to the diagonal, or equivalently np.diag(A).sum().
Hint 2
trace(A @ B) and trace(B @ A) are always equal even when A @ B and B @ A are entirely different-shaped matrices (or not even both defined) — you don't need to actually form the full product A @ B to compute its trace; summing (A * B.T) element-wise and then summing that result computes the same number more directly. Try implementing it the direct way (np.trace(A @ B)) first, then think about why the shortcut works.
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