LU Factorization Calculator
Enter a square matrix, one row per line, to factor it into lower (L) and upper (U) triangular matrices.
Separate numbers with spaces or commas. Must be square (n×n).
LU factorization writes a square matrix A as the product of a lower-triangular matrix L and an upper-triangular matrix U. Doolittle's method sets L's diagonal to 1 and solves for the other entries row by row. It is used to solve linear systems, invert matrices and find determinants efficiently.
How to Use This Calculator
Type a square matrix with one row per line, separating the numbers in a row by spaces or commas. The calculator returns the lower-triangular matrix L (with ones on its diagonal) and the upper-triangular matrix U whose product is your original matrix.
LU factorization is a workhorse of numerical linear algebra: once you have L and U, solving a system, computing a determinant, or inverting the matrix becomes fast and repeatable.
How the Calculation Works
Doolittle's method fixes the diagonal of L at 1 and then, working through the matrix, computes each entry of U across a row and each entry of L down a column using the entries already found. It is essentially Gaussian elimination, with L storing the multipliers used to zero out each column.
This calculator uses Doolittle's method without row pivoting, so it works when no pivot (diagonal U value) comes out as zero. Matrices that need row swaps require the PA = LU form with a permutation matrix.
A = LU (Doolittle): L has unit diagonal; solve U's rows and L's columns in turn
Worked Example
Factor the matrix [[4, 3], [6, 3]]. The first row of U is just the first row of A: 4 and 3. The multiplier for row 2 is 6 ÷ 4 = 1.5, so L's (2,1) entry is 1.5.
Then U's (2,2) entry is 3 − 1.5 × 3 = −1.5. So L = [[1, 0], [1.5, 1]] and U = [[4, 3], [0, −1.5]]. Multiplying L by U returns the original matrix.
Assumptions and Limitations
- Requires a square matrix (same number of rows and columns)
- Uses Doolittle's method without pivoting (L has a unit diagonal)
- Fails if a pivot is zero — such matrices need PA = LU with row swaps
- Results are rounded for display
Who Uses an LU Factorization Calculator
Linear-algebra and numerical-methods students, and engineers, use this to decompose a square matrix into lower and upper triangular factors. Run it to solve linear systems efficiently, find a determinant, or check by-hand work. Enter the matrix to get its L and U factors.
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Open calculator →Frequently Asked Questions
What is LU factorization?
It decomposes a square matrix A into a lower-triangular matrix L and an upper-triangular matrix U so that A = LU. It makes solving systems, finding determinants and inverting matrices efficient.
What is Doolittle's method?
An LU algorithm that fixes L's diagonal at 1 and computes the remaining entries of L and U row by row. It is Gaussian elimination with the elimination multipliers stored in L.
What is LU factorization used for?
Mainly for solving linear systems quickly, especially with many right-hand sides, and for computing determinants (the product of U's diagonal) and matrix inverses without redoing elimination each time.
When does LU factorization fail?
When a pivot — a diagonal entry of U — comes out zero, division is impossible without swapping rows. Then the PA = LU form with a permutation matrix P is used instead.
How do I find the determinant from LU?
For Doolittle's method, the determinant of A equals the product of the diagonal entries of U, since L has a determinant of 1. It's a quick by-product of the factorization.
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