A computational tool facilitates the decomposition of a matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition, when successful, provides a way to represent a given square matrix in terms of these two specific types of matrices. For example, a user can input a square matrix and the tool outputs the corresponding L and U matrices such that their product equals the original input.
This technique offers significant advantages in solving systems of linear equations. Instead of directly solving the system, the decomposition allows for a more efficient two-step process involving forward and backward substitution. This method proves particularly beneficial when dealing with multiple systems that share the same coefficient matrix, as the decomposition needs to be computed only once. Historically, this process became crucial in various scientific and engineering fields where solving linear systems is commonplace.
