LU Decomposition Calculator

LU decomposition expresses a square matrix A as the product L times U. The matrix L is lower triangular, meaning every entry above its main diagonal is zero. The matrix U is upper triangular, meaning every entry below its main diagonal is zero. In the Doolittle convention used here, every diagonal entry of L equals one. Multiplying the two displayed factors reconstructs the original input matrix, subject only to ordinary floating-point rounding. The factorization is built from Gaussian elimination. Each elimination multiplier that would normally be used to clear an entry below a pivot is stored in L. The evolving row-echelon matrix becomes U. This organization is valuable because it records the elimination work instead of discarding it. Once L and U are available, the same coefficient matrix can be reused to solve systems with many different right-hand sides much more efficiently than repeating elimination from the beginning. To solve A times x equals b, first solve L times y equals b by forward substitution. Then solve U times x equals y by backward substitution. Triangular systems are straightforward because each successive equation introduces only one new unknown. LU factors are also useful for computing determinants: with unit diagonal L, the determinant is the product of the diagonal entries of U. Matrix inversion and numerical simulations likewise use this structure as a basic building block. This calculator accepts compact matrix notation. Type values in each row separated by commas or spaces, and place a semicolon between rows. Every row must contain the same number of entries as the number of rows, because LU decomposition in this form requires a square matrix. Negative values and decimals are supported. Results are rounded only for display, while the elimination itself uses JavaScript double-precision arithmetic. The implementation uses unpivoted Doolittle decomposition, so each pivot encountered during elimination must be nonzero. Some perfectly invertible matrices begin with a zero pivot and require row permutation before factorization. In a full numerical library that operation is represented by a permutation matrix P in the relation P times A equals L times U. If this calculator reports a pivot error, reorder suitable rows before entering the matrix or use a pivoted solver. For well-conditioned classroom examples and matrices with usable diagonal pivots, the displayed factorization provides a clear and immediately checkable result.