Introduction
LU Decomposition, a key concept in linear algebra, is the factorization of a square matrix into the product of a lower triangular matrix and an upper triangular matrix. There are variations to this decomposition, including the pivoted LU decomposition. In this post, we'll explore both LU decomposition and its pivoted version.
Applications
LU Decomposition
LU Decomposition is frequently used to simplify the solution of a system of linear equations, such as those arising in numerical partial differential equations or optimization problems.
where \(A\) is the original matrix, \(L\) is the lower triangular matrix, and \(U\) is the upper triangular matrix.
Pivoted LU Decomposition
Pivoted LU Decomposition, on the other hand, includes a permutation matrix \(P\), which provides numerical stability - reducing the propagation of errors during the computations. It's an essential method in numerical linear algebra.
where \(P\) is the permutation matrix, \(A\) is the original matrix, \(L\) is the lower triangular matrix, and \(U\) is the upper triangular matrix.
References
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