LU Decomposition
Factor A into lower and upper triangular matrices.
Key idea
Factor A into lower and upper triangular matrices.
In depth
A = LU with L lower triangular and U upper triangular records the steps of Gaussian elimination. Solving Ax = b becomes two easy triangular solves, and L can be reused for many b.
Worked example
[2 1; 4 3] = [1 0; 2 1][2 1; 0 1].
Formula
Real-world uses
- Solving many systems with the same coefficients quickly in engineering software
Common mistake
Forgetting row swaps (pivoting), which can be needed.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is LU Decomposition?
Factor A into lower and upper triangular matrices. It is a topic at the expert level in Linear Algebra.
What is the formula for LU Decomposition?
The key formula is A = LU.
Where is LU Decomposition used in real life?
Real-world examples: Solving many systems with the same coefficients quickly in engineering software.
What should I learn before LU Decomposition?
Make sure you are comfortable with Matrix Exponential, the previous topic in Linear Algebra.
What is a common mistake with LU Decomposition?
Forgetting row swaps (pivoting), which can be needed.
How can I practise LU Decomposition?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after LU Decomposition?
Move on to Cholesky Decomposition next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.