LU Decomposition Calculator

LU decomposition factors a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, i.e., A = LU. With partial pivoting, it becomes PA = LU where P is a permutation matrix.

LU decomposition is used to efficiently solve linear systems Ax = b (via forward and back substitution), compute determinants (det(A) = product of U diagonal), and find matrix inverses.

In Doolittle's method, L has 1s on the diagonal. In Crout's method, U has 1s on the diagonal. Both produce valid decompositions; Doolittle is more commonly used.

Partial pivoting swaps rows to place the largest element in the current column on the diagonal before elimination. This prevents division by small numbers and improves numerical stability.

Not all matrices have an LU decomposition without row exchanges. Some require partial pivoting (PA = LU). Symmetric positive definite matrices always have a Cholesky decomposition, a special case of LU.

First solve Ly = Pb for y using forward substitution, then solve Ux = y for x using back substitution. This is more efficient than computing A⁻¹ and then multiplying.

The determinant equals the product of the diagonal elements of U, multiplied by (-1)^s where s is the number of row swaps (from P). det(A) = (-1)^s × product of U's diagonal.

LU decomposition requires approximately (2/3)n³ operations for an n×n matrix, making it more efficient than computing the inverse (which also requires O(n³) but with a larger constant).

Cholesky decomposition is a special case of LU where A is symmetric positive definite. It produces A = LLᵀ where L is lower triangular. It is twice as fast as general LU.

LU decomposition is defined for square matrices only. For non-square matrices, QR decomposition or singular value decomposition (SVD) are more appropriate.

LU decomposition is essentially Gaussian elimination stored in a factored form. The L matrix stores the multipliers used during elimination, and U stores the result.

To find A⁻¹, decompose A = LU, then solve A×eᵢ = eᵢ for each column of the identity matrix using forward and back substitution. This is more efficient than computing the full inverse.

LU decomposition overwrites A in place, storing L (below diagonal) and U (on and above diagonal) in the same n×n storage. This saves memory compared to storing separate matrices.

Yes, sparse LU methods exploit the sparsity pattern to reduce computation and memory. However, fill-in (new nonzeros created during elimination) can be a challenge.

LDU decomposition is A = LDU where L is unit lower triangular, D is diagonal, and U is unit upper triangular. It is a slightly more factored form of LU.

With partial pivoting, L is not purely lower triangular due to row swaps. The permutation P tracks the swaps, and the multipliers in L correspond to the permuted rows.

In block LU decomposition, the Schur complement S = D - CA⁻¹B appears when factoring a 2×2 block matrix. It plays a key role in block elimination methods.

Perform Gaussian elimination to reduce A to upper triangular form U, recording each row operation multiplier in L. L has 1s on the diagonal and the multipliers below it.

The diagonal elements of U are the pivots from Gaussian elimination. If any pivot is zero, the matrix is singular and the standard LU decomposition without pivoting fails.

Without pivoting, LU decomposition is unique when it exists. With partial pivoting, the decomposition may not be unique because different pivot choices lead to different factorizations.

The LU Decomposition Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The LU Decomposition Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.