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MathidiaMathematics Academy
Numerical algorithms
Lesson 67 of 75
Easy practice

Matrix decomposition

Objective. Factor matrices to solve repeated systems efficiently.

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

Learn: the key idea

Matrix decomposition expresses a matrix A as a product of simpler matrices, most commonly LU decomposition, which writes A = LU where L is lower triangular (with 1s on the diagonal) and U is upper triangular, obtained by recording the row-reduction steps used to turn A into U. This is useful for solving Ax=b repeatedly for different b, since Ly=b can be solved quickly by forward substitution, then Ux=y by back substitution, avoiding full elimination each time. For example, decomposing a matrix once allows many systems with the same coefficient matrix to be solved efficiently.

Worked example

Find the LU decomposition of the matrix A = [[2,1],[6,5]], where L has 1s on its diagonal.

2 … 1
Illustrated card showing 2 … 1
  1. 1To eliminate the (2,1) entry, subtract 3× row 1 from row 2, since 6/2=3
  2. 2This gives U = [[2,1],[0,2]], since row2 − 3×row1 = [6−6, 5−3] = [0,2]
  3. 3The multiplier used was 3, so L = [[1,0],[3,1]]
  4. 4Check: LU = [[1,0],[3,1]]×[[2,1],[0,2]] = [[2,1],[6,3+2]] = [[2,1],[6,5]] = A

Answer: L = [[1,0],[3,1]], U = [[2,1],[0,2]]

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Common mistake. Students write U as the row-reduced matrix but forget that L must record the multipliers used in elimination, not simply be the identity matrix. L's off-diagonal entries are the negative of the multipliers used in each elimination step (or the multipliers themselves, depending on sign convention), not zero or arbitrary values.

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