LU Decomposotion

Basically the goal of the LU decomposition is to factorize a matrix into a product of a lower triangular matrix L and an upper triangular matrix U.

Where A is the matrix we want to factorize, L is the lower triangular matrix and U is the upper triangular matrix.

The U matrix is the equivalent to the matrix we get from the Gaussian elimination method, and the L matrix is the matrix that contains the multipliers that we used to eliminate the elements below the diagonal.

So in other words we can perform the gaussian elimination method with one matrix multiplication by choosing the correct L matrix.

Just like with the Gaussian elimination method, we can use the LU decomposition to solve systems of linear equations. The resulting matrices are not unique like with the Gaussian elimination method. However, the same L can be used to solve multiple systems of equations for the same A matrix and different b vectors.

Partial Pivoting

The matrix A can be permuted with a permutation matrix P to swap rows.

Where P is a permutation matrix, A is the matrix we want to factorize, L is the lower triangular matrix and U is the upper triangular matrix.

This is called partial pivoting because we only swap rows.

What is the optimal strategy for choosing the pivot element? Maybe this should actually be written in the gaussian elimination method.

Full Pivoting

The matrix A can be permuted with a permutation matrix P to swap rows and columns.

Where P and Q are permutation matrices, A is the matrix we want to factorize, L is the lower triangular matrix and U is the upper triangular matrix.

This is called full pivoting because we swap rows and columns.

LDU Decomposition

The LU decomposition can be extended to the LDU decomposition where D is a diagonal matrix.

Where A is the matrix we want to factorize, L is the lower triangular matrix, D is the diagonal matrix and U is the upper triangular matrix.

What are the advantages?

Cholesky Decomposition

is this the same as the ldu decomposition?