LU Decomposition of Matirx
For a Matrix A, we could factor it out as A = LU
, just like we factor a number to two numbers.
Online LU Decomposition Calculator
Upper Triangular Matrix
The factor matrix U
represents the Upper Triangular Matrix
, which we're already familiar with: the matrix we've got after Gauss Elimination
.
Refer to video: LU Decomposition using Gaussian Elimination
Lower Triangular Matrix
The factor matrix L
is not hard to get as well:
All the numbers in this matrix are factor numbers
we used in each elimination step.
How to get the Lower Triangular Matrix
Refer to this video: LU Decomposition — Shortcut Method by Math is power
Solve System of equations using LU Decomposition
The final goal of learning
LU Decomposition
is to solve Linear systems.
Refer to this video: Solve a System of Linear Equations Using LU Decomposition
Assume there’s equation AX = B
as below, and we're to solve for X
:
Steps to apply the LU Decomposition
to solve the Linear System:
- Decompose LU, and represent
AX = B
asLUX = B
- Let
Y = UX
, then solveLY = B
forY
- Solve
Y = UX
forX