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LU Decomposition Calculator

Factor a square matrix into a lower L and upper U triangular matrix (Doolittle method).

Related tools

All Matrices & vectors tools →
Cholesky decomposition calculatorFactor a symmetric positive-definite matrix A into L·Lᵀ, where L is lower triangular — the Cholesky decomposition. Enter A on an editable square grid and it returns L, Lᵀ and the product L·Lᵀ; if the matrix is not symmetric or not positive-definite, it tells you which condition failed. Twice as fast as LU, Cholesky is used in optimization, Kalman filters and Monte-Carlo simulation.QR decomposition calculatorFactor a matrix A into Q·R, where Q has orthonormal columns and R is upper triangular, using the modified Gram-Schmidt process. Enter A on an editable grid (rows ≥ columns), and it returns Q, R and the product Q·R so you can check the factorization at a glance. QR is the workhorse behind least-squares regression and eigenvalue algorithms.SVD calculator (singular value decomposition)Compute the singular value decomposition A = U·Σ·Vᵀ of any matrix. The singular values are the square roots of the eigenvalues of AᵀA, found with the Jacobi eigenvalue algorithm; U and V are orthogonal. Enter A on an editable grid (up to 6×6) and it returns the singular values, U, Σ, Vᵀ and the reconstruction U·Σ·Vᵀ. SVD underpins PCA, image compression, recommender systems and the pseudo-inverse.Partial Fraction Decomposition CalculatorBreaks a rational function into simple fractions, covering distinct linear factors, repeated factors and irreducible quadratics. The setup and the linear system solved for the coefficients are both shown, since that is the part being learned.Angle Between Two Vectors Calculatorθ = arccos(a·b / (|a||b|)) in 2D or 3D, in degrees and radians, with the dot product, both magnitudes, the projections and a parallel / perpendicular / acute / obtuse verdict.Column Space Calculator (basis of Col(A), rank, pivot columns)A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.Cramer's Rule CalculatorSolves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none.Cross Product Calculatora × b in 3D with the determinant expansion shown, the magnitude |a||b|sin θ, the parallelogram and triangle areas, the unit normal and the right-hand rule.

LU Decomposition Calculator is free to use as often as you like, directly from this page. It sits under Matrices & vectors in our catalogue, alongside Cholesky decomposition calculator and QR decomposition calculator.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is LU Decomposition Calculator?

Factor a square matrix into a lower L and upper U triangular matrix (Doolittle method).

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is LU Decomposition Calculator different from Cholesky decomposition calculator?

They sit next to each other but answer different questions: Cholesky decomposition calculator is the one to open when you need it to factor a symmetric positive-definite matrix A into L·Lᵀ, where L is lower triangular — the Cholesky decomposition. Enter A on an editable square grid and it returns L, Lᵀ and the product L·Lᵀ; if the matrix is not symmetric or not positive-definite, it tells you which condition failed. Twice as fast as LU, Cholesky is used in optimization, Kalman filters and Monte-Carlo simulation. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

QR decomposition calculator is the closest one after this: Factor a matrix A into Q·R, where Q has orthonormal columns and R is upper triangular, using the modified Gram-Schmidt process. Enter A on an editable grid (rows ≥ columns), and it returns Q, R and the product Q·R so you can check the factorization at a glance. QR is the workhorse behind least-squares regression and eigenvalue algorithms.

What else is worth having open alongside it?

SVD calculator (singular value decomposition) and Partial Fraction Decomposition Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

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