matrix-operations

What problem does it solve?

This skill helps users tackle CLRS-style matrix problems by guiding factorization choices, pivoting decisions, and solving with triangular systems instead of default inverses. The core habit is to factor once, solve with triangular systems, state the rank or definiteness condition before using a shortcut, and never treat explicit inversion as the default way to solve a linear system.

Core Features & Use Cases

• Square-system factorization with LUP and triangular solves
• Reuse a factorization for many right-hand sides
• Decide whether to compute an explicit inverse
• Explain pivoting, permutation arrays, forward substitution, and back substitution
• Prove or apply symmetric positive-definite properties and Schur complements
• Derive least-squares normal equations and the full-column-rank pseudoinverse
• Recognize tridiagonal systems and spline-related problems
• Maintain CLRS tone; avoid recommending full matrix inversion as a default

Quick Start

Provide a square linear system and I will solve it using LUP and forward/back substitution, indicating whether computing an inverse is warranted.