How do you decompose a matrix into LU?
A square matrix A can be decomposed into two square matrices L and U such that A = L U where U is an upper triangular matrix formed as a result of applying the Gauss Elimination Method on A, and L is a lower triangular matrix with diagonal elements being equal to 1. ; such that A = L U.
What is LU decomposition in linear algebra?
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination.
What is advantage and disadvantage of using LU decomposition method over basic Gauss elimination method?
SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS
| Method | Advantages | Disadvantages |
|---|---|---|
| LU decomposition | Efficient if one set of linear equations is repeatedly solved with different inhomogeneous terms (e.g., in the inverse power method.) | Less efficient and more cumbersome than Gauss elimination if used only once. |
When can you not use LU decomposition?
Strictly speaking, when a row interchange is needed to get the U part of the LU, the LU Decomp doesn’t exist. In that case the matrix has a PLU decomposition, where P is a permutation matrix which is an identity matrix that’s had the row interchanges done on it.
Is the LU decomposition unique Why or why not?
LU factorizations are, as you have just discovered, not unique. Uniqueness would need some extra constraints on the form of L and U.
What is the primary drawback of using LU decomposition?
A major disadvantage is that it’s a direct method and that means very large matrices can’t be stored in memory but there other matrix decompositions which only compute matrix vector products to solve the problem.