Matrix Input
Formula Used
LU decomposition writes a square matrix as:
A = L × U.
Here, L is a lower triangular matrix.
U is an upper triangular matrix.
Doolittle Form
In Doolittle form, the diagonal entries of L are one.
The calculator computes:
U[k][j] = A[k][j] - Σ L[k][s]U[s][j]
L[i][k] = (A[i][k] - Σ L[i][s]U[s][k]) / U[k][k]
Crout Form
In Crout form, the diagonal entries of U are one.
The calculator computes:
L[i][k] = A[i][k] - Σ L[i][s]U[s][k]
U[k][j] = (A[k][j] - Σ L[k][s]U[s][j]) / L[k][k]
How to Use This Calculator
Select the matrix size first. Enter every value in the square matrix. Choose Doolittle or Crout factorization. Press the calculate button. The answer appears above the form. Review the L matrix, U matrix, determinant estimate, pivots, and residual error. Use the CSV button for spreadsheet records. Use the PDF button for reports or class notes.
Example Data Table
| Matrix | Method | Main Purpose | Expected Output |
|---|---|---|---|
| [[2, -1, -2], [-4, 6, 3], [-4, -2, 8]] | Doolittle | Classroom LU practice | L, U, pivots, determinant |
| [[4, 3], [6, 3]] | Crout | Small linear system check | Lower and upper factors |
| [[1, 2, 1], [3, 8, 1], [0, 4, 1]] | Doolittle | Numerical method review | Residual verification |
LU Matrix Factorization Guide
What This Calculator Does
This calculator separates a square matrix into two useful parts. The first part is the lower triangular matrix. The second part is the upper triangular matrix. Together, these two matrices rebuild the original matrix. This process is called LU decomposition. It is common in algebra, engineering, data science, and numerical analysis.
Why L and U Matter
Many matrix problems become easier after factorization. A large system can be split into smaller triangular systems. Triangular systems are faster to solve. They also reduce repeated work. This is useful when one coefficient matrix has many right side vectors. The method can support faster solving, determinant checks, and inverse workflows.
Advanced Options
The calculator includes two common formats. Doolittle keeps the diagonal of L equal to one. Crout keeps the diagonal of U equal to one. Both forms are valid when pivots are not zero. The tool also checks the product L × U. This confirms whether the factorization rebuilds the original matrix. A small residual means the answer is accurate.
Practical Accuracy
Matrix factorization can fail when a pivot is zero. It can also become unstable when pivots are very small. In professional software, pivoting is often added. Pivoting swaps rows to improve stability. This calculator reports a warning when direct factorization is not safe. That message helps users avoid misleading results.
Learning Benefits
The step list explains each stage. The pivot chart shows how the calculation progresses. Export options help save work. The example table gives quick starting values. Students can compare methods and see how entries change. This makes LU decomposition easier to study and apply.
Frequently Asked Questions
1. What is an L matrix?
An L matrix is a lower triangular matrix. Its entries above the main diagonal are zero. In Doolittle decomposition, its diagonal entries are usually one.
2. What is a U matrix?
A U matrix is an upper triangular matrix. Its entries below the main diagonal are zero. It works with L to recreate the original matrix.
3. What does A = L × U mean?
It means the original matrix A can be rebuilt by multiplying L and U. This factorization simplifies many linear algebra tasks.
4. Which method should I choose?
Choose Doolittle for a unit diagonal in L. Choose Crout for a unit diagonal in U. Both are useful for learning.
5. Why did the calculator show a pivot warning?
A pivot warning means a required divisor is zero or too small. Direct LU factorization may fail without row swapping.
6. Can this solve linear equations?
This page focuses on finding L and U. You can use those matrices later for forward and backward substitution.
7. What is residual error?
Residual error measures the difference between the original matrix and L × U. Smaller values mean better factorization accuracy.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet use. Use the PDF button to save a simple report of the calculation.