Matrix LU Decomposition Calculator

Welcome to the Matrix LU Decomposition Calculator, a comprehensive linear algebra tool that factors any square matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U) using Gaussian elimination with partial pivoting. Get detailed step-by-step elimination, interactive decomposition animation, and automatic verification. Ideal for students, engineers, and anyone working with systems of linear equations.

What is LU Decomposition?

LU decomposition (also called LU factorization) expresses a square matrix \(A\) as the product of two triangular matrices:

Where:

• L (Lower triangular): has 1s on the diagonal and nonzero entries only below the diagonal. These entries are the multipliers used during Gaussian elimination.
• U (Upper triangular): has nonzero entries only on and above the diagonal. This is the row echelon form of the matrix.

When partial pivoting is used (to avoid zero pivots and improve numerical stability), the factorization becomes:

Where \(P\) is a permutation matrix that records the row swaps performed during elimination.

How to Use This Calculator

1. Enter your matrix: Input a square matrix with rows on separate lines or separated by semicolons. Elements can be separated by spaces, commas, or tabs. Supports up to 8×8.
2. Set decimal precision: Choose how many decimal places (2-10) to show in results.
3. Click Decompose: The calculator performs LU factorization with partial pivoting and shows results.
4. Review results: Examine L, U, and P matrices, the animated decomposition, and step-by-step elimination details.

Solving Linear Systems with LU Decomposition

LU decomposition is particularly powerful for solving systems of linear equations \(Ax = b\). Once you have \(PA = LU\), solving becomes a two-step process:

Step 1: Forward Substitution

Solve \(Ly = Pb\) for \(y\). Since \(L\) is lower triangular, this is straightforward — start from the top equation and work down:

Step 2: Back Substitution

Solve \(Ux = y\) for \(x\). Since \(U\) is upper triangular, start from the bottom equation and work up:

Computing the Determinant

The determinant of \(A\) can be efficiently computed from the LU factors:

Where \(s\) is the number of row swaps (pivots) and \(U_{ii}\) are the diagonal entries of \(U\). Since \(\det(L) = 1\) (all diagonal entries are 1) and \(\det(P) = (-1)^s\), the formula follows from \(\det(P)\det(A) = \det(L)\det(U)\).

Why Partial Pivoting?

Without pivoting, LU decomposition fails if any pivot element is zero. Even when pivots are nonzero but small, the computed result can suffer from severe numerical errors. Partial pivoting selects the largest available pivot in each column, which:

• Prevents division by zero
• Minimizes growth of rounding errors
• Guarantees the multipliers in L satisfy \(|L_{ij}| \leq 1\)
• Ensures every nonsingular matrix can be decomposed

Applications of LU Decomposition

LU vs Other Decompositions

• LU vs QR: LU is faster (\(O(\frac{2}{3}n^3)\) vs \(O(\frac{4}{3}n^3)\)) but less numerically stable. QR is preferred for least-squares problems.
• LU vs Cholesky: Cholesky (\(A = LL^T\)) works only for symmetric positive-definite matrices but is twice as fast and more stable than general LU.
• LU vs Gaussian elimination: LU is Gaussian elimination, but the factored form L and U can be reused to solve multiple right-hand sides efficiently.

Frequently Asked Questions

What is LU decomposition?

LU decomposition (also called LU factorization) is a method that factors a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, so that A = LU (or PA = LU with partial pivoting). The L matrix has 1s on the diagonal and stores the elimination multipliers, while U is the result of Gaussian elimination.

Why is partial pivoting needed in LU decomposition?

Partial pivoting swaps rows to place the largest absolute value in the pivot position. This prevents division by zero when a pivot element is zero, and reduces numerical errors caused by dividing by small numbers. With partial pivoting, the factorization becomes PA = LU, where P is a permutation matrix recording the row swaps.

What are the applications of LU decomposition?

LU decomposition is used to solve systems of linear equations (Ax = b) efficiently, compute matrix determinants, find matrix inverses, and analyze numerical stability. It is especially efficient when solving multiple systems with the same coefficient matrix but different right-hand sides, because the factorization only needs to be done once.

How do you solve Ax = b using LU decomposition?

After computing PA = LU, solving Ax = b becomes: first solve Ly = Pb using forward substitution (easy because L is lower triangular), then solve Ux = y using back substitution (easy because U is upper triangular). This two-step process is much faster than Gaussian elimination when solving multiple systems.

Can every square matrix be LU decomposed?

Not every square matrix has an LU decomposition without pivoting. A matrix has an LU factorization if and only if all its leading principal minors are nonzero. However, with partial pivoting (PA = LU), every nonsingular square matrix can be decomposed. Singular matrices may fail if a zero pivot is encountered.

Additional Resources

• LU Decomposition - Wikipedia
• Triangular Matrix - Wikipedia
• Gaussian Elimination - Wikipedia