Symmetric eigendecomposition and thin SVD

Jacobi symmetric eigendecomposition and thin SVD for full-rank spectral computation.

eigen_full computes full eigendecompositions of symmetric N×N matrices and thin SVDs of tall matrices in pure Mojo, with zero Python imports — replacing numpy.linalg.eigh and numpy.linalg.svd (thin, tall case). It lifts the 2×2 cap of matrix_decomp.eigen_2x2 to symmetric n ≤ 200 using the cyclic Jacobi rotation method.

The modules

Module	What it does
eigen_full	full eigendecomposition of symmetric N×N matrices and thin SVD of tall matrices via cyclic Jacobi rotation, lifting the 2×2 cap of matrix_decomp.eigen_2x2 to symmetric n ≤ 200 — replacing numpy.linalg.eigh and numpy.linalg.svd (thin, tall case)

Functions

Function	What it does
eigh_symmetric(A, n, max_sweeps, tol)	Eigenvalues (sorted descending) + orthonormal eigenvectors of a symmetric n×n matrix (flat row-major). Sweeps of Givens rotations run until the off-diagonal Frobenius norm falls to tol · ‖A‖_F. Returns EighResult (eigenvalues, eigenvectors, sweeps, converged).
svd_via_eigh(A, rows, cols, max_sweeps, tol)	Thin SVD A = U · diag(σ) · V^T of a tall matrix (rows ≥ cols) via eigendecomposition of the Gram matrix A^T A. Returns SVDResult (u, singular_values, v, converged).

Guarantees and conventions

• Fully deterministic — no randomness anywhere. Eigenvector sign is fixed by making the largest-magnitude component of each vector positive.

• Eigenvalues and singular values are sorted descending; eigenvector / singular-vector k is the k-th column of the flat row-major matrix.

• Input is symmetrized as (A + A^T) / 2 before decomposition, mirroring numpy.linalg.eigh reading one triangle.

• Invalid input (n ≤ 0, n > 200, wrong flat length, or rows < cols for the SVD) returns an empty, unconverged result rather than trapping.

• Complexity is O(n³) per Jacobi sweep (typically 5–10 sweeps): sub-millisecond up to roughly n ≈ 40, tens of milliseconds at the n = 200 cap — the cap is the design envelope and bounds worst-case latency.

• SVD accuracy note: forming A^T A squares the condition number, so tiny singular values are accurate to about sqrt(machine eps) relative — fine for regression/PCA-style use. Columns of U whose σ ≤ max σ · 1e-12 are left zero (numerical null space).

Verification

The test suite pins hand-computed known values rather than snapshots:

• 2×2 [[2,1],[1,2]] recovers eigenvalues 3, 1 and eigenvectors (1,1)/√2, (1,−1)/√2 to 1e-10.

• A 3×3 constructed as Q · diag(5, 2, −1) · Q^T from a hand-chosen orthogonal Q recovers the eigenvalues and the exact Q columns to 1e-8, with eigenvector Gram deviation and reconstruction ‖A − V D V^T‖ both below 1e-8.

• The ill-conditioned 6×6 Hilbert matrix converges with orthonormality and reconstruction error below 1e-8.

• SVD: the 3×2 case recovers σ = √3 (1.7320508075688772), 1; the rank-deficient [[1,2],[2,4],[3,6]] gives σ = √70 (8.366600265340756), 0 with the null U column exactly zero; reconstruction ‖A − U Σ V^T‖ stays below 1e-8 throughout.

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