{"uid":"cap_S89BdlwcfS94vLFfWr8yP","slug":"symmetric-matrix-eigendecomposition-openblas-622f9a6e","name":"Symmetric Matrix Eigendecomposition (OpenBLAS)","description":"Computes the full eigendecomposition of a real symmetric operator and reports the signed eigenvalues, the orthonormal eigenvectors, and the inertia — how many eigenvalues are positive, negative and indistinguishable from zero at the stated tolerance. Two residuals are returned as evidence rather than as reassurance: how far the decomposition is from reproducing the input, and how far the eigenvectors are from being orthonormal.","url":"https://stat.halowerk.com/v1/oss-openblas-symmetric-spectrum-analyze?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method","bodyType"],"properties":{"type":{"const":"http"},"method":{"const":"POST"},"bodyType":{"const":"json"},"schemaUrl":{"type":"string","format":"uri"},"exampleUrl":{"type":"string","format":"uri"}},"additionalProperties":true},"output":{"type":"object","required":["type"],"properties":{"type":{"const":"json"},"schemaUrl":{"type":"string","format":"uri"},"exampleUrl":{"type":"string","format":"uri"}},"additionalProperties":true}},"additionalProperties":true},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.0015","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.0015/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.0015","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.0015","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_Jt7EmoiJVMMZL9YR0CY85","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.0015","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes the full eigendecomposition of a real symmetric matrix, returning signed eigenvalues, orthonormal eigenvectors, inertia counts, and reconstruction residuals.","exampleAgentPrompt":"Run a full eigendecomposition on this 4×4 symmetric matrix [[4,2,0,0],[2,3,1,0],[0,1,2,1],[0,0,1,5]] — I need the eigenvalues, eigenvectors, the inertia counts, and both residuals showing how accurate the decomposition is.","exampleUseCases":[{"title":"Covariance matrix spectral analysis","prompt":"I have a 6×6 sample covariance matrix and need to know its eigenvalues and eigenvectors, how many are positive versus zero, and the reconstruction error — can you run the full eigendecomposition on it? The matrix is [[5,2,1,0,0,0],[2,4,1,1,0,0],[1,1,3,0,1,0],[0,1,0,2,0,0],[0,0,1,0,1,0],[0,0,0,0,0,1]]."},{"title":"Checking positive definiteness of a kernel matrix","prompt":"I need to verify whether this 3×3 kernel matrix is positive definite — can you decompose it and tell me how many eigenvalues are positive, negative, and zero? The matrix is [[2,-1,0],[-1,2,-1],[0,-1,2]]."},{"title":"Structural stiffness matrix spectrum","prompt":"For my finite-element stiffness matrix [[10,-3,0],[-3,8,-2],[0,-2,5]], compute the full spectral decomposition — eigenvalues, eigenvectors, inertia, and both residuals so I can see how clean the factorization is."}],"resultDescription":"Returns the full set of real eigenvalues (signed, ordered), the corresponding orthonormal eigenvectors, the inertia triple (count of positive, negative, and near-zero eigenvalues at the stated tolerance), plus two numerical residuals: the Frobenius-norm distance from the reconstructed matrix to the input, and the deviation of the eigenvector matrix from exact orthonormality.","failureModes":["Matrix is not square — non-square arrays rejected with validation error","Matrix exceeds 128×128 size limit — returns schema violation error","Matrix entries outside [-1000000, 1000000] range — rejected with bounds error","Matrix is not symmetric — may proceed but results reflect the input as-is, which is numerically ill-defined for symmetric eigensolvers","Near-singular or ill-conditioned matrix — computation succeeds but residuals may be large, indicating inaccurate factorization","Empty or malformed JSON body — returns parse error"],"whenToPreferThis":"Choose this endpoint when you need a complete, verified eigendecomposition of a real symmetric matrix up to 128×128 — particularly when you need not just eigenvalues and eigenvectors but also inertia counts and objective numerical residuals to assess decomposition quality. Prefer it over general-purpose eigensolvers when working with covariance, Gram, kernel, or stiffness matrices where symmetry is guaranteed and you want OpenBLAS-backed numerical accuracy with built-in verification.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-29T00:32:44.434Z","isFirstParty":false,"canonicalSlug":"symmetric-matrix-eigendecomposition-openblas-622f9a6e"}