LU Decomposition Calculator

Decompose a matrix into Lower (L) and Upper (U) triangular matrices using Gaussian elimination.

Enter Matrix

Matrix A:

Result

Decomposition Status
A = L * U
L (Lower Triangular):
U (Upper Triangular):

Step-by-Step Gaussian Elimination

A = L * U

Understanding LU Decomposition

LU Decomposition (also known as LU Factorization) is a method of decomposing a square matrix A into a product of two matrices: a lower triangular matrix L and an upper triangular matrix U, such that A = L * U. This technique is fundamental in numerical linear algebra and is widely used for solving systems of linear equations, computing determinants, and finding matrix inverses.

How It Works

The decomposition uses Gaussian elimination to transform the original matrix into an upper triangular form (U), while recording the multipliers used during elimination in the lower triangular matrix (L). The diagonal of L consists of ones (Doolittle's method), and the entries below the diagonal represent the multipliers used during each elimination step.

The Process

  1. Start with the original matrix A.
  2. For each column, compute the multiplier for each row below the pivot: m = A[i][k] / A[k][k].
  3. Subtract m times the pivot row from the current row to create zeros below the pivot.
  4. Store the multiplier m in the corresponding position of matrix L.
  5. The resulting upper triangular matrix is U; L has 1s on its diagonal and multipliers below.

Key Properties

Lower Triangular (L)

All entries above the diagonal are zero. The diagonal entries are all 1 (in Doolittle form).

L[i][j] = 0 for i < j, L[i][i] = 1

Upper Triangular (U)

All entries below the diagonal are zero. Contains the result of Gaussian elimination.

U[i][j] = 0 for i > j

Solving Ax = b

Once A = LU, solve Ly = b (forward substitution), then Ux = y (back substitution).

Ax = LUx = L(Ux) = Ly = b

Determinant

det(A) = det(L) * det(U) = product of diagonal entries of U (since det(L) = 1).

det(A) = U[1][1] * U[2][2] * ...

Practical Applications

LU decomposition is used extensively in engineering simulations, circuit analysis, structural analysis, and computational fluid dynamics. It is more efficient than directly solving systems of equations when you need to solve the same system with multiple right-hand sides, because the decomposition only needs to be performed once.

Limitations

  • Not all matrices have an LU decomposition without row permutations (pivoting).
  • If a pivot element is zero, the decomposition fails without partial pivoting (PLU decomposition).
  • Numerical stability may require pivoting strategies for large matrices.
  • The matrix must be square for standard LU decomposition.