Cholesky Decomposition
Factor a positive definite matrix as LLᵀ.
Key idea
Factor a positive definite matrix as LLᵀ.
In depth
A symmetric positive definite matrix factors as A = LLᵀ with L lower triangular. It is about twice as fast as LU and is used in statistics and simulation.
Worked example
[4 2; 2 3] = [2 0; 1 √2][2 1; 0 √2].
Real-world uses
- Solving covariance-based equations in finance risk models and Monte Carlo simulations
Common mistake
Trying it on matrices that are not positive definite.
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 Cholesky Decomposition?
Factor a positive definite matrix as LLᵀ. It is a topic at the expert level in Linear Algebra.
Where is Cholesky Decomposition used in real life?
Real-world examples: Solving covariance-based equations in finance risk models and Monte Carlo simulations.
What should I learn before Cholesky Decomposition?
Make sure you are comfortable with LU Decomposition, the previous topic in Linear Algebra.
What is a common mistake with Cholesky Decomposition?
Trying it on matrices that are not positive definite.
How can I practise Cholesky 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 Cholesky Decomposition?
Move on to Markov Chains and Matrices 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.