Crout's Method — Definition, Formula & Examples

Crout's Method is a technique for decomposing a square matrix

A

into the product

A = LU

, where

L

is a lower triangular matrix and

U

is an upper triangular matrix with ones on its diagonal. Once you have

L

and

U

, solving a system

Ax = b

reduces to two simpler triangular solves.

Given an $n \times n$ matrix $A$ with all leading principal minors nonzero, Crout's Method computes matrices $L$ (lower triangular) and $U$ (unit upper triangular, i.e., $u_{ii} = 1$ ) such that $A = LU$ . The entries are determined column by column for $L$ and row by row for $U$ using the formulas $l_{ij} = a_{ij} - \sum_{k=1}^{j-1} l_{ik}u_{kj}$ for $i \ge j$ and $u_{ij} = \frac{1}{l_{jj}}\left(a_{ji} - \sum_{k=1}^{j-1} l_{jk}u_{ki}\right)$ for $i > j$ , with $u_{jj} = 1$ .

Key Formula

l_{ij} = a_{ij} - \sum_{k=1}^{j-1} l_{ik}\,u_{kj}, \quad u_{ji} = \frac{1}{l_{jj}}\!\left(a_{ji} - \sum_{k=1}^{j-1} l_{jk}\,u_{ki}\right)

Where:

$l_{ij}$ = Entry in row i, column j of the lower triangular matrix L
$u_{ji}$ = Entry in row j, column i of the unit upper triangular matrix U (diagonal entries are 1)
$a_{ij}$ = Entry of the original matrix A

How It Works

You process each column

j = 1, 2, \dots, n

in order. First, compute all entries

l_{ij}

in column

j

L

(rows

i \ge j

) by subtracting previously computed dot products from

a_{ij}

. Then compute all entries

u_{ji}

in row

j

U

(columns

i > j

) by dividing by the diagonal element

l_{jj}

. After obtaining

L

and

U

, solve

Ly = b

by forward substitution, then solve

Ux = y

by back substitution to get the solution

x

Worked Example

Problem: Use Crout's Method to find the LU decomposition of

A = \begin{pmatrix} 2 & 1 \\ 4 & 5 \end{pmatrix}

Step 1: Compute column 1 of L. Since j=1, there are no sums to subtract.

l_{11} = a_{11} = 2, \quad l_{21} = a_{21} = 4

Step 2: Compute row 1 of U. The diagonal entry is

u_{11} = 1

. For the off-diagonal entry, divide by

l_{11}

u_{11} = 1, \quad u_{12} = \frac{a_{12}}{l_{11}} = \frac{1}{2}

Step 3: Compute column 2 of L and set

u_{22} = 1

l_{22} = a_{22} - l_{21}\,u_{12} = 5 - 4 \cdot \tfrac{1}{2} = 3, \quad u_{22} = 1

Answer:

L = \begin{pmatrix} 2 & 0 \\ 4 & 3 \end{pmatrix}

U = \begin{pmatrix} 1 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}

. You can verify

LU = A

Why It Matters

Crout's Method is a standard algorithm taught in introductory linear algebra and numerical analysis courses. It is especially efficient when you need to solve multiple systems

Ax = b

with the same coefficient matrix but different right-hand sides, since the factorization is computed only once.

Common Mistakes

Mistake: Confusing Crout's Method with Doolittle's Method by placing the ones on the diagonal of L instead of U.

Correction: In Crout's Method, the unit diagonal belongs to U (the upper triangular factor). In Doolittle's Method, the unit diagonal belongs to L. Mixing these up produces incorrect factors.