LU Decomposition Calculator

Calculator Form

Matrix Size

Display Precision

Decomposition Mode

Use partial pivoting for better numerical stability

Enter Matrix A

Optional Right-Hand Side Vector b

Fill the vector to solve the system Ax = b. Leave all vector boxes empty to compute only the decomposition.

Example Data Table

Use this sample when testing the calculator quickly.

Row	a11	a12	a13	b
1	2	1	1	5
2	4	-6	0	-2
3	-2	7	2	9

Formula Used

Basic factorization: For a suitable square matrix, the calculator computes A = LU, where L is lower triangular and U is upper triangular.

Pivoted factorization: When partial pivoting is enabled and row swaps are needed, the relation becomes PA = LU, where P is a permutation matrix.

Doolittle formulas:

L(k,k) = 1

U(k,j) = A(k,j) - Σ[L(k,m)U(m,j)] for m = 1 to k-1

L(i,k) = (A(i,k) - Σ[L(i,m)U(m,k)]) / U(k,k) for m = 1 to k-1

Determinant from U: det(A) = sign(P) × ∏U(i,i). The sign changes with each row swap.

Solving a linear system: First solve Ly = Pb using forward substitution, then solve Ux = y using backward substitution.

How to Use This Calculator

Select the matrix size from 2 × 2 up to 6 × 6.
Enter every value of matrix A in the generated input grid.
Optionally enter vector b to solve the system Ax = b.
Choose the display precision for rounded output tables.
Keep partial pivoting enabled for improved numerical stability.
Press Compute LU Decomposition to calculate the result.
Review A, L, U, P, PA, and LU above the form.
Use the CSV and PDF buttons to export the results.

FAQs

1. What does LU decomposition mean?

LU decomposition rewrites a square matrix into a lower triangular matrix and an upper triangular matrix. This factorization simplifies solving linear systems, checking determinants, and understanding matrix structure.

2. Why is partial pivoting useful?

Partial pivoting swaps rows to place a stronger pivot on the diagonal. This reduces division by tiny numbers and usually improves stability, especially for sensitive or poorly scaled matrices.

3. Can this calculator solve Ax = b too?

Yes. Enter the optional vector b. The calculator then solves Ly = Pb and Ux = y, returning the intermediate vector y, the solution vector x, and a residual check.

4. What matrices are accepted here?

The calculator accepts square matrices from 2 × 2 through 6 × 6. Every matrix entry must be numeric. Decimals, negative values, and zero values are allowed.

5. What happens if the matrix is singular?

A singular or nearly singular matrix causes a zero or unusable pivot. When that happens, the calculator stops and shows a warning because the decomposition cannot continue reliably.

6. How is the determinant found?

Once the upper triangular matrix U is built, the determinant comes from multiplying its diagonal values. If row swaps occurred, the sign is adjusted using the permutation matrix.

7. What does the Plotly graph show?

The graph displays a heatmap of the original matrix A. It gives a quick visual sense of value distribution, sign changes, and large magnitudes across rows and columns.

8. How should I enter decimals or negative numbers?

Type standard numeric values such as 3.5, -2, 0, or 7.125. Avoid commas, symbols, or blank matrix cells. Fill every b entry only when solving a system.