Numerical Analysis 5. Matrix Factorization: LU and Cholesky

Описание: This video covers matrix factorization, focusing on LU factorization and Cholesky factorization.

LU Factorization
Definition: Factoring a non-singular square matrix A into A=LU, where L is a lower triangular matrix (with ones on the diagonal) and U is an upper triangular matrix [00:51].

Uniqueness: If it exists for a non-singular matrix, it is unique [01:40].

Process: It is closely related to Gaussian elimination; the matrix U is the result of the elimination, while L contains the factors used during the process [13:15].

Application: Used to solve linear systems Ax=b more efficiently by splitting it into two simpler triangular systems: Ly=b (forward substitution) and Ux=y (backward substitution) [20:40].

Cholesky Factorization
Definition: A special case for symmetric, positive definite matrices where A=LL T , with L being a lower triangular matrix [23:26].

Existence: A real matrix L with positive diagonal elements exists if A is symmetric and positive definite [24:13].

Efficiency: It requires approximately n 3 /6 multiplications, which is about half the computational cost of LU factorization (n 3 /3) [35:25].

Key Computational Steps
LU: Linked to Gaussian multipliers [04:01].

Cholesky: Calculated systematically term-by-term using the property that each element A ij is the dot product of rows from L and columns from L T [30:55].