The LU decomposition

The LU decomposition, or also known as lower-upper factorization, is one of the methods that solve square systems of linear equations. As its name implies, the LU factorization decomposes the A matrix into a product of two matrices: a lower triangular matrix, L, and an upper triangular matrix, U. The decomposition can be represented as follows:

Here, we can see a=l₁₁u₁₁, b=l₁₁u₁₂, and so on. A lower triangular matrix is a matrix that contains values in its lower triangle with the remaining upper triangle populated with zeros. The converse is true for an upper triangular matrix.

The definite advantage of the LU decomposition method over the Cholesky decomposition method is that it works for any square matrices. The latter only works for symmetric and positive definite matrices.

Think back to the previous example in Solving linear equations using matrices...