LU Decomposition

What is the LU decomposition method?

LU decomposition, short for Lower-Upper decomposition, is a matrix factorization technique used to break down a square matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U). It’s commonly employed to simplify solving systems of linear equations and calculating determinants.

Why is LU decomposition unique?

LU decomposition is unique because it provides a way to factorize a square matrix A into lower and upper triangular matrices (L and U) uniquely, allowing efficient solving of linear systems and determinant calculation.

How is LU decomposition calculated?

LU decomposition is calculated using Gaussian elimination, where you transform a square matrix A into lower (L) and upper (U) triangular matrices by performing row operations while keeping track of the changes in separate matrices. This process is iterative and continues until A is fully decomposed. The method with all the steps for LU decomposition is given in the article.

When LU decomposition is not possible?

LU decomposition may not be possible when the matrix A is singular (non-invertible) or when it requires pivoting for stability, but the pivot element becomes zero, causing division by zero during the decomposition process.

Are there any alternatives to LU decomposition?

Yes, alternatives to LU decomposition include Cholesky decomposition for symmetric positive definite matrices, QR decomposition for general matrices, and eigenvalue-based methods like spectral decomposition and singular value decomposition (SVD) for various matrix operations and applications.

Can LU decomposition be applied to non-square matrices?

LU decomposition is typically applied to square matrices. For rectangular matrices, QR decomposition is more commonly used. However, variations like LUP decomposition can handle rectangular matrices as well, where P is a permutation matrix.

LU decomposition of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It was introduced by Alan Turing in 1948, who also created the Turing machine.

LU decomposition method of factorizing a matrix as a product of two triangular matrices has various applications such as a solution of a system of equations, which itself is an integral part of many applications such as finding current in a circuit and solution of discrete dynamical system problems; finding the inverse of a matrix and finding the determinant of the matrix.