quantumquirkxyz Contract
- Input: problem description and inputs defined by the skill body.
- Output: Markdown artifact with completed process steps.
- Side effects: none.
- Dependencies: none.
- Stop condition: all process steps executed; artifact saved with required sections.
- Risk: low.
- Boundary: produces reasoning artifact only; no system changes.
Linear Algebra
Apply linear algebra — decompositions, solving systems, transformations — with explicit numerical stability and interpretation.
When to use
- User needs a matrix decomposition, linear system solved, or eigenproblem.
- ML, graphics, optimisation, or signal processing needs a linear-algebra backbone.
- Numerical stability of a computation is in question.
Process
- Identify problem type — linear system Ax = b, eigenvalue problem Ax = λx, SVD, least squares, PCA, or transformation.
- State matrix properties: shape, rank, symmetry, positive-definiteness, sparsity.
- Choose decomposition / method — LU (stable for well-conditioned), Cholesky (SPD), QR (least-squares), SVD (rank-deficient / ill-posed), eigendecomposition (diagonalisation).
- Compute — with explicit pivoting strategy and condition number estimate.
- Validate — residual ||Ax−b||, orthogonality of eigenvectors, singular values non-negative, backward error.
- Interpret — in the problem domain (e.g. SVD: principal components, condition number → sensitivity; eigenvalues: stability of dynamical system).
- Deliver — artifact with matrix properties, decomposition chosen, computation, residual/validation, and domain interpretation.