Hemant Anufriyev

This 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.Basically, the LU decomposition method comes in handy whenever it is possible to model the problem to be solved into matrix form. Conversion to the matrix form and solving with triangular matrices makes it easy to do calculations in the process of finding the solution.

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.

For A = , we have L = and U = ; such that A = L U.

Here value of l21 , u11 etc can be compared and found.

Gauss Elimination MethodAccording to the Gauss Elimination method:

Steps for LU Decomposition:

Example:Solve the following system of equations using LU Decomposition method:

Solution: Here, we have

A = and such that A X = C.

Now, we first consider and convert it to row echelon form using Gauss Elimination Method.

So, by doing

(1)

(2)

we get

Now, by doing

(3)

we get

(Remember to always keep ‘ – ‘ sign in between, replace ‘ + ‘ sign by two ‘ – ‘ signs)

Hence, we get L = and U =

(notice that in L matrix, is from (1), is from (2) and is from (3))

Now, we assume Z and solve L Z = C.

So, we have

Solving, we get , and .

Now, we solve U X = Z

Therefore, we get ,

Thus, the solution to the given system of linear equations is , , and hence the matrix X =

Exercise:In the LU decomposition of the matrix

, if the diagonal elements of U are both 1, then the lower diagonal entry l22 of L is (GATE CS 2015)(A) 4(B) 5(C) 6(D) 7For Solution, see https://www.geeksforgeeks.org/gate-gate-cs-2015-set-1-question-28/

This article is compiled by Nishant Arora. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

1. Harvard University
2. University of California, Irvine
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4. École Polytechnique
5. Michigan State University
6. 6 California Institute of the Arts
7. Hong Kong University of Science and Technology
8. UCL (University College London)

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