LUdecomposed

LUdecomposed is a term that refers to a matrix that has undergone LU decomposition. LU decomposition, also known as LU factorization, is a method of matrix decomposition that breaks down a given square matrix into the product of two matrices: a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition is particularly useful in numerical linear algebra for solving systems of linear equations, finding the determinant of a matrix, and inverting matrices.

The process of LU decomposition involves transforming the original matrix into an upper triangular matrix through a series of elementary row operations, which are recorded in the lower triangular matrix. The lower triangular matrix L typically contains the multipliers used in the row operations, with ones on its diagonal. The upper triangular matrix U is the result of applying these operations to the original matrix.

A LUdecomposed matrix, therefore, is no longer the original matrix but its factored form, L * U. This representation can significantly simplify computational tasks. For instance, solving Ax = b becomes solving Ly = b and then Ux = y, which are much faster to compute than directly solving Ax = b using methods like Gaussian elimination. The determinant of the original matrix can also be easily found by multiplying the diagonal elements of the U matrix, as the determinant of L is 1. While not all matrices can be decomposed into LU form without pivoting, the concept of a LUdecomposed matrix highlights the utility of this factorization technique.