Concept Architecture
Concept
Theoretically, LU Decomposition is a matrix factorisation method that decomposes a square matrix into the product of a lower triangular matrix and an upper triangular matrix. It is a fundamental technique in numerical linear algebra and scientific computing that enables efficient solution of systems of linear equations, matrix inversion and determinant calculation. The concept is widely used in health economics within optimisation algorithms, numerical simulation and decision-analytic modelling.
Mathematically, LU Decomposition expresses a matrix as the product of lower and upper triangular matrices. For many matrices, a permutation matrix is incorporated to account for row exchanges required for numerical stability, resulting in the more general decomposition PA = LU. The triangular matrices are obtained using Gaussian elimination while preserving computational efficiency and numerical accuracy.
In practice, LU Decomposition is extensively applied in health economic modelling to solve large systems of equations arising in Markov models, optimisation routines, calibration algorithms and probabilistic sensitivity analyses. Compared with repeatedly applying Gaussian elimination, LU factorisation allows multiple systems sharing the same coefficient matrix to be solved efficiently.
Purpose
Used to factorise matrices efficiently, solve systems of linear equations, perform matrix inversion and support numerical optimisation and simulation within health economics.
Mathematical Formulae
Primary Formula
A = LU
where:
- A = square matrix
- L = lower triangular matrix
- U = upper triangular matrix
Supporting Formulae
General Form with Pivoting
PA = LU
where:
- P = permutation matrix
Matrix Solution
Ax = b
LUx = b
Ly = b
Ux = y
Related Mathematical Methods
- Gaussian elimination
- Cholesky decomposition
- QR decomposition
- Matrix inversion
- Forward substitution
- Back substitution
Example
Consider the matrix
A =
[ \begin{bmatrix} 4 & 3 \ 6 & 3 \end{bmatrix} ]
Its LU decomposition is
L =
[ \begin{bmatrix} 1 & 0 \ 1.5 & 1 \end{bmatrix} ]
U =
[ \begin{bmatrix} 4 & 3 \ 0 & -1.5 \end{bmatrix} ]
Verification:
LU =
[ \begin{bmatrix} 1 & 0 \ 1.5 & 1 \end{bmatrix} \begin{bmatrix} 4 & 3 \ 0 & -1.5 \end{bmatrix}
\begin{bmatrix} 4 & 3 \ 6 & 3 \end{bmatrix} = A
The decomposition therefore exactly reconstructs the original matrix.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(L2:M3,N2:O3) | Verify LU matrix reconstruction. |
| MINVERSE | =MINVERSE(A2:B3) | Matrix inversion using LU-derived solutions. |
| MDETERM | =MDETERM(A2:B3) | Calculate determinants using decomposed matrices. |
| TRANSPOSE | =TRANSPOSE(A2:B3) | Support related matrix operations. |
| LET/LAMBDA | Custom recursive formulas | Implement LU decomposition algorithms in Excel. |
VBA (Optional)
Automate LU decomposition for solving large systems of linear equations, optimisation routines, calibration methods and decision-analytic health economic models.
Sources
- Golub GH, Van Loan CF. Matrix Computations.
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Trefethen LN, Bau D. Numerical Linear Algebra.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (3)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is LU decomposition?
A matrix factorisation method that expresses a square matrix as the product of a lower triangular matrix and an upper triangular matrix.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What does LU decomposition express a matrix as?
LU decomposition expresses a square matrix as the product of a lower triangular matrix and an upper triangular matrix. The original matrix is split into these two triangular factors, one with entries on and below the diagonal and one on and above it. Writing a square matrix as a lower times an upper triangular matrix is what LU decomposition does By splitting a matrix into two triangular factors that are easy to solve against, the factorisation turns a difficult system into two simple ones, which is why it underpins much practical linear algebra.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What kind of matrix does LU decomposition apply to?
LU decomposition applies to a square matrix, since it is a factorisation that expresses such a matrix as the product of a lower triangular matrix and an upper triangular matrix. The requirement of squareness fits the factorisation into two triangular factors of the same size. Applying to a square matrix is part of what LU decomposition is By splitting a matrix into two triangular factors that are easy to solve against, the factorisation turns a difficult system into two simple ones, which is why it underpins much practical linear algebra.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What are the two factors in LU decomposition?
The two factors in LU decomposition are a lower triangular matrix and an upper triangular matrix, whose product reconstructs the original square matrix. The lower factor holds its non-zero entries on and below the diagonal and the upper factor on and above it. These two triangular factors are what LU decomposition produces By splitting a matrix into two triangular factors that are easy to solve against, the factorisation turns a difficult system into two simple ones, which is why it underpins much practical linear algebra.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Why is the triangular form useful in LU decomposition?
The triangular form is useful in LU decomposition because a lower or upper triangular matrix is easy to solve with directly, so expressing a square matrix as their product turns a harder problem into two simpler triangular ones. This is why factorising a matrix into triangular factors is worthwhile. The convenience of triangular factors underlies the value of LU decomposition By splitting a matrix into two triangular factors that are easy to solve against, the factorisation turns a difficult system into two simple ones, which is why it underpins much practical linear algebra.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does LU decomposition relate to matrix inversion?
LU decomposition expresses a square matrix as the product of a lower and an upper triangular matrix, while matrix inversion is the process of finding a matrix that, multiplied by the original, gives the identity. LU decomposition provides an efficient route to inversion, since the triangular factors are easy to invert or to solve against. The two are connected, since the factorisation supports computing an inverse By splitting a matrix into two triangular factors that are easy to solve against, the factorisation turns a difficult system into two simple ones, which is why it underpins much practical linear algebra.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Apr 2026
Content version: 1.0.0
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