Concept Architecture
Concept
Theoretically, Singular Value Decomposition (SVD) is a matrix factorisation method that decomposes any real or complex matrix into the product of two orthogonal matrices and a diagonal matrix containing non-negative singular values. It is one of the most fundamental results in numerical linear algebra and provides the basis for dimensionality reduction, matrix approximation, regression, principal component analysis and numerical optimisation. The concept is widely applied in health economics for multivariate analysis, machine learning, probabilistic modelling and high-dimensional data analysis.
Mathematically, Singular Value Decomposition expresses a matrix as the product of left singular vectors, singular values and right singular vectors. The singular values are the square roots of the eigenvalues of the matrix cross-products and quantify the amount of variation or information contained in each orthogonal dimension. The decomposition exists for every matrix regardless of whether it is square, symmetric or singular.
In practice, Singular Value Decomposition is used extensively in health economics to reduce the dimensionality of large datasets, stabilise regression estimation, solve ill-conditioned systems, perform latent semantic analyses, estimate low-rank matrix approximations and support machine learning algorithms. It also forms the computational foundation for many implementations of principal component analysis and related multivariate techniques.
Purpose
Used to factorise matrices, reduce dimensionality, solve ill-conditioned numerical problems, estimate low-rank approximations and support multivariate statistical and machine learning methods within health economics.
Mathematical Formulae
Primary Formula
A = U�V?
where:
- A = original matrix
- U = orthogonal matrix of left singular vectors
- � = diagonal matrix of singular values
- V? = transpose of the orthogonal matrix of right singular vectors
Supporting Formulae
A?A = V��V?
AA? = U��U?
Singular Value
�? = �??
where ?? is an eigenvalue of A?A.
Related Mathematical Methods
- Eigenvalue decomposition
- Principal component analysis
- QR decomposition
- LU decomposition
- Cholesky decomposition
- Matrix approximation
Example
Consider the matrix
A =
[ \begin{bmatrix} 3 & 0 \ 0 & 2 \end{bmatrix} ]
Its Singular Value Decomposition is
U =
[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]
� =
[ \begin{bmatrix} 3 & 0 \ 0 & 2 \end{bmatrix} ]
V? =
[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]
Verification:
U�V? = A
The decomposition therefore exactly reconstructs the original matrix while separating its orthogonal directions and singular values.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(U2:V3,MMULT(W2:X3,Y2:Z3)) | Verify SVD matrix reconstruction. |
| TRANSPOSE | =TRANSPOSE(A2:B3) | Calculate matrix transposes required for SVD algorithms. |
| MINVERSE | =MINVERSE(A2:B3) | Support related numerical matrix computations. |
| MDETERM | =MDETERM(A2:B3) | Evaluate matrix properties during decomposition workflows. |
| LET/LAMBDA | Custom numerical algorithms | Implement iterative SVD procedures in Excel. |
VBA (Optional)
Automate Singular Value Decomposition for dimensionality reduction, principal component analysis, regression estimation, machine learning and numerical optimisation within health economic models.
Sources
- Golub GH, Van Loan CF. Matrix Computations.
- Trefethen LN, Bau D. Numerical Linear Algebra.
- Strang G. Linear Algebra and Its Applications.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- Jolliffe IT. Principal Component Analysis.
- 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 (5)
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 singular value decomposition?
A matrix factorisation method that separates a matrix into orthogonal components and singular values representing its principal directions and scaling.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What does singular value decomposition separate a matrix into?
Singular value decomposition separates a matrix into orthogonal components and singular values, the singular values representing the matrix's principal directions and scaling. The orthogonal components describe directions and the singular values describe how much the matrix stretches along them. Splitting a matrix into these orthogonal parts and singular values is what singular value decomposition does By exposing the directions along which a matrix acts most strongly and by how much, the factorisation underpins tasks such as data compression, dimensionality reduction and finding low-rank approximations.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What do the singular values represent in singular value decomposition?
In singular value decomposition, the singular values represent the matrix's principal directions and scaling, indicating how strongly the matrix acts along each direction picked out by its orthogonal components. Larger singular values mark directions of greater stretching. Capturing this scaling along principal directions is the role of the singular values By exposing the directions along which a matrix acts most strongly and by how much, the factorisation underpins tasks such as data compression, dimensionality reduction and finding low-rank approximations.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What kind of components does singular value decomposition produce?
Singular value decomposition produces orthogonal components alongside singular values, the orthogonal parts describing directions that are mutually perpendicular. These structured components, combined with the singular values, express the matrix's principal directions and scaling. Producing orthogonal components and singular values is what singular value decomposition yields By exposing the directions along which a matrix acts most strongly and by how much, the factorisation underpins tasks such as data compression, dimensionality reduction and finding low-rank approximations.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What kind of method is singular value decomposition?
Singular value decomposition is a matrix factorisation method, separating a matrix into orthogonal components and singular values that reveal its principal directions and scaling. Rather than solving one specific problem, it rewrites the matrix into structured parts. Being a factorisation into orthogonal components and singular values is what singular value decomposition is By exposing the directions along which a matrix acts most strongly and by how much, the factorisation underpins tasks such as data compression, dimensionality reduction and finding low-rank approximations.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does singular value decomposition relate to QR decomposition?
Singular value decomposition separates a matrix into orthogonal components and singular values representing its principal directions and scaling, while QR decomposition expresses a matrix as the product of an orthogonal matrix and an upper triangular matrix. Both are matrix factorisation methods involving orthogonal factors, but they reveal different structure. They are connected as factorisations that rewrite a matrix into components suited to different purposes By exposing the directions along which a matrix acts most strongly and by how much, the factorisation underpins tasks such as data compression, dimensionality reduction and finding low-rank approximations.
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: 16 Jun 2026
Content version: 1.0.0
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- CS-LA-MF-004
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