mat_eigh — MATH `linalg` op

• Data kinds: matrix → table

• Call: import mathops; mathops.mat_eigh(a) (or opsmath.get("mat_eigh"))

Usage

Eigen-decomposition of a symmetric matrix (LAPACK `syevd`).

Returns `(w, V): eigenvalues w` in ascending order (all real —

guaranteed by symmetry) and orthonormal eigenvectors as the columns of

`V (A @ V[:, i] == w[i] * V[:, i]`).

Symmetric input only, verified: `max|A - A.T| above 1e-10` of the

matrix scale raises `ValueError`. This is deliberate fail-closing of two

traps at once — a symmetric solver fed a non-symmetric matrix silently

reads one triangle and returns a *plausible wrong* answer, and a general

matrix has complex eigenvalues this real-valued API cannot even represent.

For a covariance / Hessian / Gram matrix (the metrology cases) symmetry

holds by construction; symmetrise explicitly (`(A + A.T) / 2`) if yours

is symmetric-up-to-noise.

Sign trap (honest): each eigenvector is defined only up to sign, and

eigenvectors of a *repeated* eigenvalue only up to rotation in that

subspace. Compare `|v·w|` or subspaces, never raw columns.

HALCON: `eigenvalues_symmetric_matrix`.

Family-wide input contract (fail-closed)

Every mathops op validates its input before computing (nothing slips through silently):

• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.

• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.

• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).

• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).

• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).

Detailed usage guide

• math_metrology family guide

Background guides (the physics and conventions behind this op)

• blas_threads_and_memory — 行列分解が遅い理由の知識 — BLAS スレッド・キャッシュ・メモリ配置

References (sample data, literature)

• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).

• Operator provenance and references — the sources of the research/methods this op family came from.

• The canonical algorithm (author, year) and its uses are named in the family usage guide above.

Runnable examples (verified samples that actually call this op)

• math_metrology — py -3.11 examples/math_metrology.py

Ops the type connects to (they accept `table` as input)

—

Same category (`linalg`)

mat_solve · mat_lstsq · mat_svd · mat_pinv · mat_cond

*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

mat_eigh — MATH linalg op