matrix-theory-specialist skill

Overview

This skill is an advanced matrix theory specialist focused on spectral theory, matrix factorizations, and numerical linear algebra. It summarizes theory, recommends algorithms, and highlights numerical stability and implementation concerns. Note: the current content is a legacy template and requires research updates; treat outputs as advisory until confidence is upgraded.

How this skill works

The skill inspects matrix properties (eigenvalues, singular values, norms), derives or cites factorization methods (LU, QR, SVD, Cholesky, Schur) and evaluates numerical behavior (conditioning, stability, backward/forward error). It offers step-by-step algorithmic choices for computation, suggests preconditioning and scaling strategies, and flags situations where symbolic reasoning or high-precision arithmetic is recommended. Outputs include proofs sketches, complexity estimates, and implementation notes for common numeric libraries.

When to use it

• Selecting algorithms for eigenvalue or singular value computations
• Choosing factorizations for solving linear systems or least-squares problems
• Assessing conditioning, stability, and error sources in numeric workflows
• Designing preconditioners and iterative solvers for large sparse matrices
• Explaining theoretical results in spectral theory or matrix perturbation

Best practices

• Always check matrix conditioning before using direct solvers; prefer stable factorizations like QR or SVD for ill-conditioned problems
• Use pivoting and equilibration (row/column scaling) to reduce numerical instability
• Favor backward-stable algorithms and quantify error with forward/backward analysis
• For large sparse problems, prefer iterative methods with tailored preconditioners over dense factorizations
• Document assumptions (symmetry, definiteness, sparsity) to select appropriate methods and guarantee correctness

Example use cases

• Deciding between dense SVD and randomized SVD for low-rank approximation in machine learning
• Deriving error bounds for eigenvalue perturbations under matrix perturbation models
• Designing a preconditioner pipeline for conjugate gradient on a discretized PDE
• Choosing a factorization and implementation strategy for stable least-squares in analytics
• Explaining why Cholesky fails on nearly indefinite matrices and suggesting remedies

FAQ