Glossary

2.5 Eigenvalues and eigenvectors

8 min read•august 21, 2024

and are powerful tools in mathematical economics, helping analyze complex systems and predict long-term behavior. These concepts provide insights into matrix transformations, system dynamics, and equilibrium states, forming the foundation for advanced economic analysis techniques.

In this section, we'll explore the definition and properties of eigenvalues and eigenvectors, their geometric interpretation, and applications in economics. We'll also dive into , spectral decomposition, and , showcasing how these tools simplify complex economic models and enhance our understanding of economic phenomena.

Definition of eigenvalues

  • Eigenvalues play a crucial role in linear algebra and mathematical economics, providing insights into matrix transformations and system dynamics
  • In economic models, eigenvalues help analyze long-term behavior, stability, and equilibrium states of complex systems
  • Understanding eigenvalues forms the foundation for advanced economic analysis techniques, including and dynamic equilibrium systems

Characteristic equation

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  • Derived from the matrix equation (A−λI)v=0(A - \lambda I)v = 0, where A represents the square matrix, λ denotes the eigenvalue, and v stands for the eigenvector
  • Obtained by setting the of (A−λI)(A - \lambda I) to zero, resulting in a polynomial equation in λ
  • Roots of the correspond to the eigenvalues of the matrix
  • Degree of the equals the size of the square matrix
  • Solving the characteristic equation reveals important properties of the linear transformation represented by the matrix

Geometric interpretation

  • Eigenvalues represent scaling factors applied to eigenvectors during linear transformations
  • Visualized as the amount of stretching or compression along the direction of the corresponding eigenvector
  • Positive indicate stretching, while negative real eigenvalues signify reflection and stretching
  • Complex eigenvalues represent rotations combined with scaling in the complex plane
  • Eigenvalues of 1 or -1 indicate preservation of length or reflection without scaling, respectively

Properties of eigenvalues

  • Eigenvalues provide crucial information about matrix behavior in economic models and linear systems
  • Understanding eigenvalue properties helps economists analyze system stability, growth rates, and long-term equilibrium states
  • Eigenvalue analysis forms the basis for advanced techniques in dynamic economic modeling and policy analysis

Trace and determinant

  • of a matrix equals the sum of its eigenvalues
  • Determinant of a matrix equals the product of its eigenvalues
  • For a 2x2 matrix, eigenvalues can be calculated using the quadratic formula λ=12(tr(A)±tr(A)2−4det(A))\lambda = \frac{1}{2}(tr(A) \pm \sqrt{tr(A)^2 - 4det(A)})
  • Trace and determinant provide quick insights into matrix properties without full eigenvalue computation
  • Useful in economic models to assess system stability and growth rates

Algebraic vs geometric multiplicity

  • refers to the number of times an eigenvalue appears as a root of the characteristic equation
  • denotes the dimension of the eigenspace associated with an eigenvalue
  • Algebraic multiplicity always greater than or equal to geometric multiplicity
  • When algebraic multiplicity exceeds geometric multiplicity, the matrix is defective
  • have incomplete sets of eigenvectors, requiring generalized eigenvectors for full analysis

Eigenvectors and eigenspaces

  • Eigenvectors and form the backbone of matrix decomposition techniques in economic modeling
  • Understanding these concepts allows economists to simplify complex systems and analyze their behavior over time
  • Eigenvector analysis provides insights into the principal components of economic systems and their interactions

Eigenvector computation

  • Solve the homogeneous system (A−λI)v=0(A - \lambda I)v = 0 for each eigenvalue λ
  • Reduced row echelon form (RREF) used to find the general solution of the homogeneous system
  • Normalize eigenvectors to unit length for standardization and easier interpretation
  • Eigenvectors corresponding to distinct eigenvalues are linearly independent
  • In economic models, eigenvectors represent the directions of principal components or key factors driving system behavior

Basis of eigenspace

  • Eigenspace consists of all eigenvectors associated with a particular eigenvalue, including the zero vector
  • formed by linearly independent eigenvectors corresponding to the same eigenvalue
  • Dimension of eigenspace equals the geometric multiplicity of the eigenvalue
  • Eigenspace basis useful for analyzing subspaces invariant under the linear transformation
  • In economic applications, eigenspace analysis helps identify key structural relationships and invariant properties of systems

Diagonalization

  • Diagonalization simplifies complex economic systems by transforming them into a more manageable form
  • This technique allows economists to analyze long-term behavior, stability, and equilibrium states of
  • Diagonalization forms the foundation for many advanced economic modeling techniques and policy analysis tools

Conditions for diagonalization

  • Matrix must have a full set of linearly independent eigenvectors
  • Number of linearly independent eigenvectors must equal the size of the square matrix
  • Algebraic multiplicity of each eigenvalue must equal its geometric multiplicity
  • Diagonalizable matrices have n linearly independent eigenvectors for an n x n matrix
  • Non-diagonalizable matrices require more advanced techniques ()

Diagonalization process

  • Construct matrix P with eigenvectors as columns
  • Compute P^(-1), the inverse of the eigenvector matrix
  • Diagonal matrix D contains eigenvalues on the main diagonal
  • Diagonalization equation A=PDP−1A = PDP^{-1}
  • Powers of A easily computed using An=PDnP−1A^n = PD^nP^{-1}, simplifying analysis of dynamic systems
  • Diagonalization facilitates solving systems of differential equations in economic models

Applications in economics

  • Eigenvalue and eigenvector analysis form the cornerstone of many advanced economic modeling techniques
  • These mathematical tools allow economists to analyze complex systems, predict long-term behavior, and develop effective policies
  • Understanding eigenvalue applications provides insights into economic structures, growth patterns, and equilibrium states

Input-output analysis

  • Developed by Wassily Leontief to study interdependencies between economic sectors
  • Uses eigenvalue analysis to determine the equilibrium state of an economy
  • Eigenvalues of the input-output matrix indicate the overall growth or decline of the economy
  • Eigenvectors represent the relative importance of different sectors in the economic structure
  • Stability of the input-output model assessed through eigenvalue analysis of the Leontief inverse matrix

Dynamic systems modeling

  • Eigenvalues determine the stability and long-term behavior of dynamic economic systems
  • Negative real parts of eigenvalues indicate stable systems, while positive real parts suggest instability
  • Complex eigenvalues with negative real parts result in damped oscillations in economic variables
  • Eigenvectors show the directions of movement in phase space for different economic variables
  • Used in growth models, business cycle analysis, and macroeconomic policy evaluation

Spectral decomposition

  • Spectral decomposition provides a powerful tool for analyzing economic data and simplifying complex models
  • This technique allows economists to identify principal components and key drivers in economic systems
  • Understanding spectral decomposition enhances the ability to interpret and predict economic phenomena

Spectral theorem

  • Applies to normal matrices (matrices that commute with their conjugate transpose)
  • States that normal matrices can be unitarily diagonalized
  • Decomposition takes the form A=UDU∗A = UDU^*, where U is unitary and D is diagonal
  • Eigenvalues appear on the diagonal of D, and columns of U are the corresponding eigenvectors
  • Spectral decomposition used in factor analysis and principal component analysis in econometrics

Symmetric matrices

  • All eigenvalues of symmetric matrices are real
  • Eigenvectors of distinct eigenvalues are orthogonal to each other
  • Symmetric matrices always diagonalizable with orthogonal eigenvectors
  • Spectral decomposition of symmetric matrices takes the form A=QDQTA = QDQ^T, where Q is orthogonal
  • Covariance matrices in econometrics are symmetric, making spectral decomposition particularly useful in data analysis

Eigenvalue algorithms

  • Efficient eigenvalue computation algorithms are essential for analyzing large-scale economic models and datasets
  • These algorithms enable economists to handle complex systems and perform advanced statistical analyses
  • Understanding eigenvalue algorithms helps in selecting appropriate tools for specific economic modeling tasks

Power method

  • Iterative algorithm to find the dominant eigenvalue and corresponding eigenvector
  • Starts with a random vector and repeatedly multiplies it by the matrix
  • Converges to the eigenvector associated with the largest absolute eigenvalue
  • Convergence rate depends on the ratio of the two largest eigenvalues
  • Used in Google's PageRank algorithm and in economic models to find long-term equilibrium states

QR algorithm

  • More general method for computing all eigenvalues and eigenvectors of a matrix
  • Iteratively performs QR decomposition and updates the matrix
  • Converges to an upper triangular matrix with eigenvalues on the diagonal
  • Efficient for dense matrices and widely used in numerical linear algebra software
  • Applied in factor analysis and principal component analysis in econometrics

Generalized eigenvectors

  • Generalized eigenvectors extend eigenvalue analysis to defective matrices common in economic models
  • This concept allows economists to analyze systems that lack a full set of linearly independent eigenvectors
  • Understanding generalized eigenvectors enables more comprehensive analysis of complex economic structures

Jordan canonical form

  • Generalizes diagonalization for matrices with incomplete sets of eigenvectors
  • Decomposes a matrix into block diagonal form, with Jordan blocks on the diagonal
  • Each Jordan block corresponds to an eigenvalue and contains 1's on the superdiagonal
  • Jordan form J=P−1APJ = P^{-1}AP, where P contains generalized eigenvectors
  • Used to analyze the behavior of defective systems in economic models

Defective matrices

  • Matrices with algebraic multiplicity greater than geometric multiplicity for some eigenvalues
  • Cannot be diagonalized due to insufficient linearly independent eigenvectors
  • Require generalized eigenvectors to form a complete basis
  • Generalized eigenvectors satisfy (A−λI)kv=0(A - \lambda I)^k v = 0 for some k > 1
  • Analysis of defective matrices crucial in some economic models with degenerate equilibria or structural singularities

Eigenvalues of special matrices

  • Special matrix structures often arise in economic models due to underlying system properties
  • Understanding the eigenvalue characteristics of these matrices simplifies analysis and provides insights into model behavior
  • Knowledge of special matrix eigenvalues helps economists interpret model results and design efficient solution methods

Triangular matrices

  • Eigenvalues of appear on the main diagonal
  • Upper and lower triangular matrices have the same eigenvalue properties
  • Simplifies eigenvalue computation for these matrix structures
  • Triangular matrices often arise in economic models with hierarchical or sequential relationships
  • Useful in analyzing supply chain models and multi-stage production processes

Companion matrices

  • Associated with characteristic polynomials of other matrices
  • Eigenvalues of companion matrix match roots of the corresponding polynomial
  • Used to convert higher-order difference or differential equations to first-order systems
  • Facilitates analysis of dynamic economic models with lagged variables
  • Companion matrix eigenvalues provide insights into system stability and oscillatory behavior

Stability analysis

  • Stability analysis forms a crucial part of economic modeling, particularly in dynamic systems and policy evaluation
  • Eigenvalue-based stability analysis helps economists predict long-term behavior and design effective interventions
  • Understanding stability concepts allows for more robust economic forecasting and policy recommendations

Eigenvalues and stability

  • Negative real parts of eigenvalues indicate stable systems
  • Positive real parts suggest instability and exponential growth
  • Zero real parts result in neutral stability or constant oscillations
  • Magnitude of eigenvalues determines the rate of convergence or divergence
  • Complex eigenvalues with negative real parts produce damped oscillations in economic variables

Phase portraits

  • Graphical representation of system dynamics in state space
  • Eigenvectors determine the direction of movement along principal axes
  • Eigenvalues dictate the nature of movement (attraction, repulsion, or rotation)
  • Saddle points occur when eigenvalues have opposite signs
  • Phase portraits help visualize equilibrium points, cycles, and trajectories in economic models

Key Terms to Review (31)

Algebraic Multiplicity: Algebraic multiplicity is defined as the number of times a particular eigenvalue appears as a root of the characteristic polynomial of a matrix. This concept is crucial when discussing eigenvalues and eigenvectors because it helps to determine how many linearly independent eigenvectors correspond to each eigenvalue. Essentially, algebraic multiplicity gives insight into the structure of the matrix and its behavior in transformation.
Basis of Eigenspace: The basis of eigenspace refers to a set of linearly independent eigenvectors associated with a specific eigenvalue of a matrix. This basis spans the eigenspace, which is the subspace formed by all eigenvectors corresponding to that eigenvalue, including the zero vector. Understanding the basis of eigenspace is crucial because it helps in analyzing the structure of linear transformations and solving systems of linear equations efficiently.
Cayley-Hamilton Theorem: The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic polynomial. This means that if you take a matrix and find its characteristic polynomial, plugging the matrix back into this polynomial will yield the zero matrix. This theorem connects deeply with eigenvalues and eigenvectors, as it allows for the determination of matrix properties using these concepts.
Characteristic Equation: A characteristic equation is a polynomial equation derived from a matrix or a differential equation that helps identify the eigenvalues of the matrix or system. It plays a crucial role in understanding the behavior of linear transformations, determining stability in systems of ordinary differential equations, and analyzing eigenvalues and eigenvectors. By solving the characteristic equation, you can find the values that allow for significant insights into system dynamics.
Characteristic Polynomial: The characteristic polynomial is a polynomial derived from a square matrix that encodes important information about the matrix, particularly its eigenvalues. It is obtained by taking the determinant of the matrix subtracted by a scalar multiple of the identity matrix, and its roots correspond to the eigenvalues of the original matrix. Understanding this polynomial is crucial for solving systems of linear equations and analyzing linear transformations in mathematical economics.
Companion Matrices: A companion matrix is a special type of square matrix that represents a linear transformation related to a polynomial. It is constructed from the coefficients of a polynomial, where the first column contains the coefficients of the polynomial in reverse order, and the rest of the entries are arranged in a way that reflects the polynomial's structure. Companion matrices are significant because they provide a convenient way to analyze the eigenvalues and eigenvectors associated with a polynomial, connecting directly to concepts of linear transformations and characteristic polynomials.
Complex Eigenvectors: Complex eigenvectors are vectors associated with complex eigenvalues of a matrix, which arise when the characteristic polynomial has non-real roots. These vectors indicate the directions along which a linear transformation, represented by the matrix, acts by simply stretching or compressing, often leading to rotations in the complex plane. They play a crucial role in understanding the behavior of systems described by linear transformations, particularly in contexts where oscillations or waves are involved.
Defective Matrices: A defective matrix is a square matrix that does not have a complete set of linearly independent eigenvectors. This occurs when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity. Understanding defective matrices is crucial because they can impact the stability and behavior of systems described by differential equations, particularly in relation to eigenvalues and eigenvectors.
Determinant: A determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix, such as whether it is invertible. The determinant plays a crucial role in various mathematical concepts, especially in solving systems of linear equations and understanding transformations represented by matrices. The value of the determinant can reveal information about the volume scaling factor of the linear transformation associated with the matrix.
Diagonalization: Diagonalization is the process of transforming a matrix into a diagonal form, where all non-diagonal elements are zero, which makes certain calculations easier. This process is deeply connected to eigenvalues and eigenvectors, as diagonalizable matrices can be expressed in terms of these eigenvalues, simplifying the understanding of their properties and applications in various fields such as economics and engineering.
Dynamic Systems: Dynamic systems are mathematical models that represent how variables change over time, capturing the essence of systems in motion. These models are particularly important in analyzing processes where current states influence future states, often using differential equations or difference equations. The behavior of dynamic systems can provide insights into stability, growth, and oscillatory patterns, which are essential for understanding economic phenomena.
Eigenspaces: An eigenspace is a subspace associated with a particular eigenvalue of a matrix, consisting of all eigenvectors corresponding to that eigenvalue along with the zero vector. This concept highlights how certain transformations can stretch or compress vectors in specific directions without changing their direction, showcasing the geometric interpretation of eigenvalues and eigenvectors. Eigenspaces play a crucial role in understanding linear transformations and their effects on vector spaces.
Eigenvalues: Eigenvalues are special numbers associated with a square matrix that provide important information about the linear transformations represented by that matrix. When a matrix acts on a vector, the eigenvalues tell us how much the eigenvector is stretched or shrunk and in which direction it is pointing. These concepts are foundational in understanding vector spaces, linear transformations, and dynamic systems, allowing us to analyze stability and behavior over time.
Eigenvectors: Eigenvectors are special vectors associated with a square matrix that, when that matrix is multiplied by the eigenvector, result in a vector that is a scalar multiple of the original eigenvector. This property highlights the significance of eigenvectors in understanding linear transformations, where they indicate directions that remain unchanged under the transformation. Eigenvectors, along with their corresponding eigenvalues, play a vital role in various applications including stability analysis and differential equations.
Geometric Multiplicity: Geometric multiplicity refers to the number of linearly independent eigenvectors associated with a particular eigenvalue of a matrix. It provides insight into the dimensionality of the eigenspace corresponding to that eigenvalue, indicating how many directions in which a transformation can stretch or compress vectors. This concept is crucial for understanding the behavior of linear transformations represented by matrices, as it directly influences properties like diagonalizability and stability.
Input-Output Models: Input-output models are mathematical representations used to analyze the relationships between different sectors of an economy by depicting how the output from one sector becomes an input to another. These models help to understand the interdependencies among industries, showcasing how changes in one sector can impact others. They utilize matrices to represent these transactions and can be used for forecasting economic impacts, analyzing trade-offs, and studying the effects of policy changes.
Jordan Canonical Form: Jordan Canonical Form is a special type of matrix representation that simplifies linear transformations, making it easier to understand their structure, particularly when dealing with eigenvalues and eigenvectors. This form consists of Jordan blocks along the diagonal, which correspond to the eigenvalues of the matrix. By transforming a matrix into its Jordan form, one can effectively analyze its properties, especially in cases where the matrix does not have a full set of linearly independent eigenvectors.
Linear Independence: Linear independence is a concept in linear algebra where a set of vectors is considered independent if no vector in the set can be expressed as a linear combination of the others. This means that the only way to combine these vectors to get the zero vector is by multiplying all of them by zero. Understanding linear independence is crucial for analyzing vector spaces and the behavior of eigenvalues and eigenvectors, as it helps determine the dimensionality of these spaces and whether a set of vectors spans a space without redundancy.
Markov Chains: Markov chains are mathematical systems that transition from one state to another within a finite or countable set of states, where the probability of each transition depends only on the current state and not on the previous states. This property is known as the Markov property, making these chains particularly useful in various fields like economics, statistics, and computer science for modeling random processes over time. Their connection to eigenvalues and eigenvectors comes into play when analyzing the long-term behavior and stability of these systems through the transition matrix.
Power Method: The power method is an iterative algorithm used to find the dominant eigenvalue and its corresponding eigenvector of a square matrix. This technique is especially useful when dealing with large matrices, as it focuses on the largest eigenvalue, allowing for efficient computation. The power method works by repeatedly multiplying a vector by the matrix and normalizing it, which causes the vector to converge towards the eigenvector associated with the largest eigenvalue.
QR Algorithm: The QR algorithm is a numerical method used to compute the eigenvalues and eigenvectors of a matrix by decomposing it into a product of an orthogonal matrix (Q) and an upper triangular matrix (R). This technique is particularly effective for finding eigenvalues, as it iteratively refines the approximation and can converge to the actual values efficiently, making it a key tool in linear algebra and computational mathematics.
Real eigenvalues: Real eigenvalues are scalar values associated with a square matrix that indicate the factor by which the corresponding eigenvector is scaled during a linear transformation. They are crucial in understanding the behavior of linear transformations and can provide insights into the stability and dynamic properties of systems represented by matrices. When a matrix has real eigenvalues, it often implies that the transformation does not rotate or reflect vectors, but simply scales them in the direction of their corresponding eigenvectors.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra that states that every normal matrix can be diagonalized by a unitary matrix. This means that for any normal matrix, there exists an orthonormal basis of eigenvectors corresponding to its eigenvalues. The theorem not only helps in understanding the structure of matrices but also has significant applications in various fields such as quantum mechanics and principal component analysis.
Square Matrices: A square matrix is a matrix with the same number of rows and columns, which means it has an equal dimension of m x m. This structure allows for unique mathematical operations, such as finding eigenvalues and eigenvectors, since these operations are specifically defined for square matrices. The properties and characteristics of square matrices are crucial for understanding linear transformations and their applications in various fields.
Stability analysis: Stability analysis is a method used to determine the stability of equilibrium points in dynamic systems, particularly in economics. This concept is essential for understanding how systems respond to changes and whether they will return to equilibrium or diverge away from it when subjected to perturbations. Analyzing the stability of a system helps in predicting long-term behavior and assessing the impact of various shocks or changes in parameters.
Stochastic matrix: A stochastic matrix is a square matrix used to describe the transitions of a Markov chain, where each of its rows represents a probability distribution. Each entry in the matrix is non-negative and the sum of the entries in each row equals one, reflecting the total probability of transitioning from one state to another. This concept is essential in understanding dynamic systems where outcomes are partly random and partly determined by current states.
Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose, meaning that the elements across the diagonal are mirrored. This property results in the condition that for any element at position (i, j), it holds that the value is the same as that at position (j, i). Symmetric matrices play an important role in various mathematical operations and are particularly significant when discussing eigenvalues and eigenvectors, as they often lead to real eigenvalues and orthogonal eigenvectors.
Trace: The trace of a square matrix is defined as the sum of its diagonal elements. This concept is significant in linear algebra as it relates to eigenvalues, providing insights into the properties of matrices and their transformations. The trace is also useful in various applications, such as understanding the behavior of linear transformations and solving systems of equations.
Triangular Matrices: Triangular matrices are square matrices where all the entries either above or below the main diagonal are zero. There are two types: upper triangular matrices, where all entries below the diagonal are zero, and lower triangular matrices, where all entries above the diagonal are zero. These matrices simplify many mathematical operations, particularly in relation to eigenvalues and eigenvectors, as they make solving systems of equations more straightforward.
V (eigenvector): An eigenvector is a non-zero vector that, when a linear transformation is applied to it via a matrix, results in a scalar multiple of itself. This means that the direction of the eigenvector remains unchanged even though its magnitude may be scaled by a corresponding eigenvalue. Eigenvectors are fundamental in understanding linear transformations and play a crucial role in various applications such as stability analysis, optimization, and econometrics.
λ (lambda): In the context of linear algebra, λ (lambda) represents an eigenvalue, which is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation. When a matrix acts on an eigenvector, the result is a scalar multiple of that eigenvector, where the scalar is the eigenvalue. This relationship is fundamental in understanding systems of equations and stability in various fields including economics.
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