For AI agents: use /llms.txt for the Nakafa content index.
For positive definite matrices, there is a special property that makes decomposition much simpler. LU decomposition can be performed without using a permutation matrix because Gaussian elimination can proceed without row swapping, and all pivot elements generated are guaranteed to be positive.
This means we obtain factorization in the form , where the diagonal elements of are positive pivot elements for all diagonal indices.
Since , we also have:
where is a matrix whose main diagonal is normalized to , and is a diagonal matrix:
Since LU decomposition without is unique, then:
If we define:
then .
Positive definite matrices allow for Cholesky decomposition:
where is a regular lower triangular matrix. This matrix can be computed using the Cholesky algorithm.
The computation of matrix is performed with:
based on the relationship . The following algorithm produces the Cholesky factor.
Given a positive definite matrix .
For :
for .
After running this algorithm, we will obtain the Cholesky factor which is a lower triangular matrix:
The Cholesky algorithm for computing the Cholesky factor from requires:
arithmetic operations.
This is half the number of operations required to compute LU decomposition, because the use of symmetry allows us to perform computations without row swapping in a different order.
