IU Decomposition Preview

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Factor a square matrix \(A\) into \(A = LU\) (lower/upper triangular). Optionally use partial pivoting so that \(PA = LU\). If you provide \(\mathbf{b}\), the tool also solves \(A\mathbf{x}=\mathbf{b}\) using forward and back substitution.

Size (n×n)

Pivoting

Display decimals

Tolerance

Steps

Solve \(A\mathbf{x}=\mathbf{b}\) (enter \(\mathbf{b}\))

Play animates elimination snapshots once and stops automatically. Drag to pan • wheel to zoom • double-click to reset.

Matrix \(A\)

3×3

Vector \(\mathbf{b}\)

3×1

Ready

Lower triangular \(L\)

Unit diagonal

Upper triangular \(U\)

After elimination

Permutation \(P\)

Shown only when pivoting is enabled

Elimination animation (building L and U)

Shows the current \(U\) during elimination; pivot cell and target row are highlighted.

U pivot pivot row target row

Enter A (and optionally b) and click “Calculate”.

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Theory

What LU decomposition means

The LU decomposition factors a square matrix \(A\in\mathbb{R}^{n\times n}\) into the product of a lower triangular matrix \(L\) and an upper triangular matrix \(U\). In the common “Doolittle” convention used by this calculator, \(L\) has ones on its diagonal: \[ A = LU,\quad L=\begin{bmatrix} 1 & 0 & \cdots & 0\\ \ell_{21} & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ \ell_{n1} & \ell_{n2} & \cdots & 1 \end{bmatrix}, \quad U=\begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ 0 & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & u_{nn} \end{bmatrix}. \] Triangular matrices are easy to work with because solving systems \(L\mathbf{y}=\mathbf{b}\) or \(U\mathbf{x}=\mathbf{y}\) can be done by simple substitution.

LU is closely related to Gaussian elimination. When you eliminate entries below a pivot, you are effectively building \(U\), and the multipliers used in elimination become the entries of \(L\). This is why LU is often described as “Gaussian elimination written as a factorization.”

Why pivoting is sometimes needed

If a pivot \(u_{kk}\) is zero (or extremely small), dividing by it to form multipliers is impossible (or numerically unstable). A standard fix is partial pivoting, which swaps the current row \(k\) with a row below it that has a larger absolute value in the pivot column. Row swaps can be represented by a permutation matrix \(P\). With pivoting, the factorization becomes: \[ PA = LU. \] Here, \(P\) encodes the row swaps performed during elimination.

Pivoting improves numerical stability and allows LU to work on many matrices where a no-pivot algorithm would fail. The calculator lets you choose between “No pivoting” (aiming for \(A=LU\)) and “Partial pivoting” (aiming for \(PA=LU\)).

Solving Ax=b using LU

The main practical reason to compute LU is to solve linear systems efficiently. If you want \(\mathbf{x}\) such that \(A\mathbf{x}=\mathbf{b}\), and you have \(A=LU\), then: \[ LU\mathbf{x}=\mathbf{b}. \] Let \(\mathbf{y}=U\mathbf{x}\). Then you solve two simpler systems:

Forward substitution: Solve \(L\mathbf{y}=\mathbf{b}\) from top to bottom.
Back substitution: Solve \(U\mathbf{x}=\mathbf{y}\) from bottom to top.

If pivoting is used and \(PA=LU\), then the correct right-hand side is \(P\mathbf{b}\): \[ PA\mathbf{x}=P\mathbf{b}\quad\Rightarrow\quad LU\mathbf{x}=P\mathbf{b}. \] The calculator shows \(P\mathbf{b}\), then computes \(\mathbf{y}\) and \(\mathbf{x}\) via substitution.

A key advantage: if you need to solve the same matrix \(A\) with many different \(\mathbf{b}\) vectors, you compute LU once, then each new solve only uses substitution (which is fast).

Interpreting the steps and the animation

During elimination, for each pivot position \(k\), the tool forms multipliers \(\ell_{ik}=u_{ik}/u_{kk}\) for rows \(i>k\). The multiplier \(\ell_{ik}\) is stored in \(L\), and the row is updated in \(U\) as: \[ \text{Row}_i \leftarrow \text{Row}_i - \ell_{ik}\,\text{Row}_k. \] The Play animation shows snapshots of the evolving \(U\) and the multipliers being recorded in \(L\). Highlighting indicates the pivot cell and the target row currently being eliminated.

The residual check \(\max|PA-LU|\) (or \(\max|A-LU|\) when pivoting is off) is a quick way to confirm the factorization: the closer it is to 0, the more accurate the decomposition is at the chosen display precision.

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