matrix-operations

Matrix Operations

Overview

This skill turns matrix-operation questions into method choices with visible hypotheses. The core habit is: 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.

Shared CLRS Conventions

Use the parent clrs skill for display-math-only formatting, theorem-precondition discipline, and CLRS answer tone. In this chapter skill, keep tables verbal; place every matrix equation, cost expression, or symbolic condition in display LaTeX blocks outside the table.

When to Use

Use this skill for:

• solving square linear systems by Gaussian elimination, LU decomposition, or LUP decomposition;
• reusing a factorization for many right-hand sides;
• deciding whether to compute an explicit inverse;
• explaining pivoting, permutation arrays, forward substitution, or back substitution;
• proving facts about symmetric positive-definite matrices and Schur complements;
• deriving least-squares normal equations and the full-column-rank pseudoinverse;
• recognizing tridiagonal linear systems and spline-style systems from the chapter problems.

Do not use this skill as the main guide for Strassen-style matrix multiplication; route those questions to divide-and-conquer unless the prompt is about its equivalence with inversion.

First Decision

Situation	Choose	Why
Square nonsingular system	LUP decomposition, then triangular solves	Pivoting avoids zero pivots and is the practical chapter default
Many right-hand sides for the same matrix	Reuse one LUP factorization	Each additional solve is quadratic; explicit inverse is not needed
User asks for the inverse itself	Solve one system per identity-column after LUP	The inverse is the deliverable, not a shortcut for solving
Matrix is known symmetric positive-definite	LU without pivoting is theoretically safe	Leading submatrices and Schur complements remain positive-definite
Least-squares with more observations than basis functions	Build normal equations if full column rank holds	The normal-equation matrix is symmetric positive-definite under that hypothesis
Rectangular, rank-deficient, or numerically delicate prompt	State the chapter limit first	Do not invent unprovided QR, SVD, or rank-revealing claims unless asked beyond the chapter