LU Decomposition, or LU factorization, is a method in linear algebra to factorize a matrix \(A\) into the product of two other matrices: a lower triangular matrix \(L\) and an upper triangular matrix \(U\).

\(A = LU\)

The matrix \(L\) has 1s on its main diagonal, and the matrix \(U\) has the pivots on its main diagonal.

How it's Solved:

This calculator uses the Doolittle algorithm. It iteratively calculates the elements of L and U based on the following formulas:

\( u_{ij} = a_{ij} - \sum_{k=1}^{i-1} l_{ik}u_{kj} \)

\( l_{ij} = \frac{1}{u_{jj}} \left( a_{ij} - \sum_{k=1}^{j-1} l_{ik}u_{kj} \right) \)

This process is essential for efficiently solving systems of linear equations, finding determinants, and inverting matrices.