LU Decomposition Calculator

Understanding 3×3 LU Decomposition

This LU decomposition calculator expresses a square matrix $A$ as the product of a lower triangular matrix $L$ and an upper triangular matrix $U$. Mathematically, we write $A=LU$. Triangular matrices are useful in numerical work because they simplify solving linear systems and computing determinants. By decomposing $A$ into $L$ and $U$, we can solve $Ax=b$ using two easier steps: first solve $Ly=b$ via forward substitution, then solve $Ux=y$ via backward substitution. This factorization is central to many numerical algorithms.

For the matrices accepted by this calculator, the Doolittle factorization proceeds without row exchanges. The entries of $L$ have ones on the diagonal, and its entries below the diagonal store the elimination multipliers. The matrix $U$ is the upper triangular result of elimination. If you carry out the row operations by hand, each multiplier used to eliminate an entry below the diagonal becomes an element of $L$. The triangular shapes then make the associated systems especially straightforward to solve.

The Doolittle Method for a 3×3 Matrix

This calculator uses the Doolittle method, which fixes the diagonal of $L$ at one while placing the diagonal and upper entries in $U$. The factorization is built from the entries of the 3×3 matrix $A$, written as $a_{ij}$. We determine the first column of $L$ by setting $l_{21}=\frac{a_{21}}{a_{11}}$ and $l_{31}=\frac{a_{31}}{a_{11}}$. Then we compute the first row of $U$ as $u_{1j}=a_{1j}$. The remaining entries come from subtracting products of already known $L$ and $U$ entries from the matching entries of $A$. Each step is chosen so that multiplying $L$ by $U$ reconstructs $A$.

The Doolittle procedure used here makes LU decomposition's relationship to Gaussian elimination explicit. Eliminating entries below the diagonal transforms $A$ into the upper triangular matrix $U$, while the corresponding row-elimination multipliers are retained in $L$. Thus the factorization preserves both the final triangular matrix and the operations that produced it.

Applications of LU Decomposition in Linear Systems

LU decomposition is particularly valuable when several linear systems use the same coefficient matrix. After factoring $A$ once, a different right-hand side $b$ can be handled by solving $Ly=b$ and then $Ux=y$. For a no-pivoting factorization, the determinant is also the product of the diagonal entries of $U$, because $L$ has a unit diagonal. Numerical linear algebra libraries use LU-based techniques in tasks including differential-equation discretization and control-system calculations.

LU factors can also reveal a problem with a matrix before a linear system is solved. A zero pivot prevents this no-pivoting procedure; a very small pivot can make floating-point results sensitive to rounding. More general library routines address many such cases with row pivoting, yielding $PA=LU$, where $P$ is a permutation matrix. This page instead shows the unpivoted Doolittle form so the elimination arithmetic remains visible.

How to use: the LU Decomposition Calculator

To factor a 3×3 matrix with this LU calculator, enter its nine entries row by row: a11 through a13, then a21 through a23, and finally a31 through a33. Press Compute LU to apply the no-pivoting Doolittle algorithm and display the $L$ and $U$ matrices rounded to four decimal places. If either of the first two pivots is zero, the calculator reports that the decomposition failed. Because partial pivoting is not included, choose a nonsingular matrix whose required pivots are nonzero when using this tool for a worked factorization.

Introduction: Why LU Decomposition Matters

LU decomposition turns the repeated work of Gaussian elimination on a 3×3 coefficient matrix into reusable triangular factors. In a finite-difference model, circuit equations, or a graphics transformation, those factors allow forward and backward substitution instead of repeating a full elimination for every new right-hand side. Studying the split into $L$ and $U$ also helps connect hand elimination to the matrix routines used in engineering and scientific software.

LU Decomposition Example Walkthrough

This LU decomposition example begins with the matrix $\left[\begin{array}{ccc}2& 3& 1\\ 4& 7& 2\\ 6& 18& 5\end{array}\right]$. Using the Doolittle method, the first pivot is $2$. The multipliers for the first column become $l_{21}=\frac{4}{2}=2$ and $l_{31}=\frac{6}{2}=3$. After eliminating the first column from the remaining rows, the second pivot is nonzero, so the process can continue. This example shows how the multipliers accumulate below the diagonal in $L$ while $U$ holds the eliminated upper-triangular form.

For this 3×3 LU decomposition, each entered matrix entry can affect later pivots and multipliers, so manual arithmetic is easy to miscopy. Use the displayed factors to check the elimination sequence, and verify that the selected matrix does not create a zero first or second pivot. Trying matrices from coursework or a model is a practical way to observe how changing one entry changes both factors.

Beyond Basic LU Decomposition

After learning the unit-diagonal Doolittle form used by this calculator, you can compare other triangular factorizations. Crout's method puts the diagonal entries in $L$ rather than $U$. For a symmetric positive-definite matrix, Cholesky decomposition writes $A=L{L}^T$. Large numerical problems may instead use sparse or block LU methods to exploit matrix structure.

Experimenting with the calculator's lower and upper triangular outputs builds intuition for how elimination is encoded in matrix form. That perspective is useful when moving on to pivoting, iterative refinement, preconditioning, and numerical stability. LU decomposition is therefore both a hand-calculation technique and a building block in computational linear algebra.

Formula: how this LU factorization is built

This calculator computes the Doolittle factorization $A=LU$ for the nine entered matrix entries. It sets the diagonal entries of $L$ to one, uses the first and second pivots to calculate the below-diagonal multipliers, and obtains $U$ by subtracting the associated elimination products. Enter every aij value as an ordinary numeric matrix entry; the form has no mixed units or percentage fields.

Limitations and assumptions of this LU calculator

This LU decomposition calculator handles only a 3×3 matrix and the Doolittle method without row pivoting. Its output is reliable only when the entered values are numeric and the first two computed pivots are nonzero; matrices requiring row exchanges need a pivoted LU method instead. Rounded display values are convenient for inspection, but retain sufficient precision in source calculations when matrix conditioning matters.