Concept Architecture
Concept
Theoretically, an Eigenvector is a non-zero vector whose direction remains unchanged when a linear transformation represented by a square matrix is applied, although its magnitude may be scaled by a corresponding eigenvalue. Eigenvectors are fundamental objects in linear algebra and matrix theory and underpin numerous statistical, econometric and health economic methods, including principal component analysis, factor analysis, Markov modelling and dynamic systems analysis.
Mathematically, an eigenvector is determined by solving the matrix equation associated with a given eigenvalue. Once an eigenvalue has been obtained from the characteristic equation of the matrix, the corresponding eigenvector is found by solving the associated homogeneous system of linear equations. Eigenvectors identify the principal directions of variation or system behaviour represented by the matrix.
In practice, eigenvectors are widely used in health economics for principal component analysis, factor analysis, transition probability modelling, dimensionality reduction and numerical optimisation. Within principal component analysis, eigenvectors define the directions of the principal components, while in Markov models they describe long-term system behaviour and steady-state characteristics.
Purpose
Used to identify the principal directions of linear transformations, support multivariate statistical methods, analyse dynamic systems and facilitate matrix decomposition within health economics and related disciplines.
Mathematical Formulae
Primary Formula
Av = ?v
where:
- A = square matrix
- v = eigenvector
- ? = corresponding eigenvalue
Supporting Formulae
(A ? ?I)v = 0
Characteristic Equation
det(A ? ?I) = 0
Related Mathematical Methods
- Eigenvalue analysis
- Matrix decomposition
- Principal component analysis
- Factor analysis
- Singular value decomposition
- Markov chain analysis
Example
Consider the matrix
A =
[ \begin{bmatrix} 4 & 0 \ 0 & 2 \end{bmatrix} ]
The eigenvalues are:
?? = 4
?? = 2
For ?? = 4, one corresponding eigenvector is
v? =
[ \begin{bmatrix} 1 \ 0 \end{bmatrix} ]
For ?? = 2, one corresponding eigenvector is
[ \begin{bmatrix} 0 \ 1 \end{bmatrix} ]
These vectors indicate the directions that remain unchanged under the matrix transformation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(A2:B3,D2:D3) | Verify eigenvector relationships through matrix multiplication. |
| MUNIT | =MUNIT(2) | Generate identity matrices when solving eigenvector equations. |
| MINVERSE | =MINVERSE(A2:B3) | Support matrix algebra used in eigenvector calculations. |
| MDETERM | =MDETERM(A2:B3) | Evaluate determinants during characteristic equation analysis. |
| LET/LAMBDA | Custom matrix algorithms | Implement numerical eigenvector estimation procedures. |
VBA (Optional)
Automate computation of eigenvectors using matrix decomposition algorithms for principal component analysis, factor analysis, Markov modelling and multivariate statistical applications.
Sources
- Strang G. Linear Algebra and Its Applications.
- Golub GH, Van Loan CF. Matrix Computations.
- Trefethen LN, Bau D. Numerical Linear Algebra.
- 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 an eigenvector?
A non-zero vector whose direction remains unchanged when transformed by a matrix, although its magnitude may be scaled.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What is special about an eigenvector under a matrix transformation?
An eigenvector is special because its direction remains unchanged when it is transformed by a matrix, even though its magnitude may be scaled. Most vectors are rotated into a new direction by a matrix, but an eigenvector keeps pointing the same way, only stretched or shrunk. This preservation of direction under the transformation is what defines an eigenvector Because a matrix leaves an eigenvector's direction fixed and only scales it, such vectors pick out the axes along which the matrix acts most simply, exposing structure in how it transforms space.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
Why must an eigenvector be non-zero?
An eigenvector must be non-zero because the zero vector has no direction to preserve, and the defining property of an eigenvector is that its direction remains unchanged when transformed by a matrix. Allowing the zero vector would make the condition trivial, since it stays put under any transformation. Restricting eigenvectors to non-zero vectors is therefore part of the definition Because a matrix leaves an eigenvector's direction fixed and only scales it, such vectors pick out the axes along which the matrix acts most simply, exposing structure in how it transforms space.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What may change for an eigenvector under transformation?
For an eigenvector, its magnitude may change under transformation, since the definition allows the vector to be scaled while its direction remains unchanged. The matrix may stretch or shrink the eigenvector, but it does not turn it to point elsewhere. This possible change in magnitude, alongside the fixed direction, is characteristic of how a matrix acts on an eigenvector Because a matrix leaves an eigenvector's direction fixed and only scales it, such vectors pick out the axes along which the matrix acts most simply, exposing structure in how it transforms space.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
What does an eigenvector reveal about a matrix?
An eigenvector reveals a direction that the matrix leaves unchanged, only scaling the vector along it, since an eigenvector is a non-zero vector whose direction remains unchanged when transformed. Such directions expose how the matrix acts, picking out axes along which its effect is a simple stretch or shrink. Identifying an eigenvector therefore uncovers structure in how the matrix transforms space Because a matrix leaves an eigenvector's direction fixed and only scales it, such vectors pick out the axes along which the matrix acts most simply, exposing structure in how it transforms space.
Source: Golub GH, Van Loan CF. Matrix Computations. 4th ed. Johns Hopkins University Press; 2013.
How does an eigenvector relate to an eigenvalue?
An eigenvector is a non-zero vector whose direction remains unchanged when transformed by a matrix, although its magnitude may be scaled, while an eigenvalue is the scalar indicating how much that scaling is. The eigenvalue is the factor by which the matrix stretches or shrinks its eigenvector. The two are paired, since each eigenvector has an associated eigenvalue giving the amount of scaling Because a matrix leaves an eigenvector's direction fixed and only scales it, such vectors pick out the axes along which the matrix acts most simply, exposing structure in how it transforms space.
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: 2 Apr 2026
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/eigenvector
- Term code
- CS-LA-EA-002
Stable URI · Machine-readable · Resolvable · CC BY 4.0