Solve CLRS-style matrix problems using LUP decomposition and triangular substitution.
npx skills add https://github.com/Arcadi4/nerdy --skill matrix-operations
Please help me install this Agent Skill. Skill: matrix-operations Source: https://github.com/Arcadi4/nerdy/tree/main/clrs/matrix-operations Command: npx skills add https://github.com/Arcadi4/nerdy --skill matrix-operations
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.
Provide a square linear system and I will solve it using LUP and forward/back substitution, indicating whether computing an inverse is warranted.
High-intent search queries and answers about installing and using this skill.
To solve a square linear system with LUP decomposition, factor the matrix into lower, upper, and permutation matrices, then use forward and back substitution with permutation arrays. This approach reuses one factorization for multiple right-hand sides and avoids explicit matrix inversion unless specifically required.
Compute an explicit matrix inverse only when the application specifically demands the inverse itself, such as analyzing sensitivity or specific theoretical proofs. For solving linear systems or handling multiple right-hand sides, factor once with LUP and use forward and back substitution instead of blind inversion.
Factor the square system matrix once using LUP decomposition and store the resulting lower, upper, and permutation arrays. Then, for each new right-hand side, apply forward and back substitution to reuse the single factorization and efficiently solve the triangular systems.
Apply Schur complements and symmetric positive-definite properties by verifying the definiteness condition, then partitioning the block matrix to isolate the complement. This lets you derive solutions for block matrices and prove properties without calculating a full explicit inverse.
Derive least-squares normal equations by multiplying the transpose of the design matrix by the system, then solve the resulting square system using LUP decomposition. For full-column-rank matrices, compute the pseudoinverse by applying the factorization to the normal equations.
Solve tridiagonal systems by recognizing the banded structure and applying specialized triangular solves that exploit the zero-pattern, avoiding general LUP overhead. Recognize spline-related problems by identifying the specific tridiagonal structure in the interpolation matrix.
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