LU Factorization Calculator

Calculator for performing LU factorization without pivoting of a matrix.

The definition for LU factorization (aka LU decomposition) used here is

Let $A$ be an $m \times n$ matrix. A presentation $A = LU$ , in which $L is$ $m \times m$ unit lower triangular and $U$ is $n \times n$ upper triangular, is called an $LU$ factorization of $A$ .

Usage notes

Input a matrix in the form of a 2D array in JavaScript

E.g., to find the factorization of $\begin{bmatrix}6 & 12 \cr 18 & 20 \end{bmatrix}$ input [[6,12],[18,20]]

The "Details" disclosure widget (displayed below the results) shows the matrices used to create the factorization. That is, $A = LU = [L_0L_*^{-1}][L_*\mathscr{U}]$.
If the matrix has infinitely many factorizations, the elements that can be any complex number will be shown as $\alpha_1,\alpha_2,...,\alpha_n$, where $n$ is the total number of such elements.
If the matrix has no possible factorization, then an approximation will be made. Elements of the inputted matrix that prevent the factorization from materializing will be increased by $\varepsilon$, and factorization of this modified matrix will be shown.
This calculator uses the approach to obtain all LU factorizations of a matrix described in

Froilán M. Dopico, Charles R. Johnson, and Juan M. Molera, Multiple LU factorizations of a singular matrix, Linear Algebra and its Applications. 419 (2006), no. 1, 24–36, DOI 10.1016/j.laa.2006.03.043.

Concerning when a matrix has a single, infinitely many, or no factorization, see

Ly Jacky Nhiayi and Tuyetdong Phan-Yamada, Examining Possible LU Decompositions, North American GeoGebra Journal. 9 (2021), no. 1, 1–7.