Skip to main content
MathidiaMathematics Academy
Scientific computing
Lesson 56 of 75
Introduction

Matrix computation

Objective. Choose stable factorizations for large systems.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Matrix computation chooses numerically stable factorizations, such as LU with partial pivoting or Cholesky for symmetric positive-definite systems, to solve large linear systems accurately despite rounding error. LU decomposition writes A = LU (lower and upper triangular factors), allowing fast forward and back substitution for multiple right-hand sides, while partial pivoting swaps rows to avoid dividing by small pivots that amplify error. For example, factorising [[2,1],[4,3]] gives multiplier 4/2 = 2, so row2 − 2·row1 gives U = [[2,1],[0,1]] and L = [[1,0],[2,1]]. Cholesky, A = LLᵀ, is preferred when A is symmetric positive-definite because it halves the computational cost.

Worked example

In a matrix computation problem, find the LU decomposition of A = [[1,2],[3,8]], giving the multiplier used to eliminate the entry below the first pivot.

1 … 2
Illustrated card showing 1 … 2
  1. 1Identify the pivot as the (1,1) entry of A, which is 1.
  2. 2Compute the multiplier needed to eliminate the (2,1) entry 3, as m = 3/1.
  3. 3Simplify m = 3/1 = 3, so row2 − 3·row1 is used to form U.
  4. 4This multiplier 3 becomes the (2,1) entry of L in A = LU.

Answer: 3

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common mistake is performing LU factorization without pivoting on a matrix with a small or zero pivot, which causes huge rounding errors or division by zero; partial pivoting should always be used for general matrices.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard