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 as LUX = B

• Let Y = UX, then solve LY = B for Y

• Solve Y = UX for X