Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Numerical Analysis → Numerical Linear Algebra ↓


Eigenvalue Computations


Eigenvalue computations play a key role in numerical linear algebra and are important in many applications ranging from solving systems of differential equations to synchronizing complex systems. In short, eigenvalues and eigenvectors provide a way to understand the intrinsic properties of matrices, which are fundamental in mathematical modeling and computation.

Introduction to eigenvalues and eigenvectors

In simple terms, the eigenvalue of a matrix is a scalar that indicates how much the associated eigenvector is stretched or contracted during the linear transformation represented by the matrix.

The fundamental equation defining the eigenvalue λ and the eigenvector v is:

Av = λv

where A is a square matrix, v is a non-zero vector, and λ is a scalar. A matrix can have multiple eigenvalues, and each eigenvalue can have one or more corresponding eigenvectors.

Understanding eigenvalue problems

The eigenvalue problem arises when we wish to determine the values (eigenvalues) of λ for which there exists a nonzero vector v such that the matrix equation Av = λv holds.

For this equation to be valid for non-zero v, the matrix (A - λI) must not be invertible. This gives the characteristic equation:

det(A - λI) = 0

Here, I is the identity matrix of the same size as A, and the determinant being zero indicates that the matrix (A - λI) is not invertible.

Calculating eigenvalues: basic methods

Power iteration

Power iteration is the simplest algorithm for finding the largest (in absolute value) eigenvalue of a matrix A and its corresponding eigenvectors. Here is how you can implement it:

def power_iteration(A, num_simulations):
    b_k = np.random.rand(A.shape[1])
    for _ in range(num_simulations):
        # Calculate the matrix-by-vector product Ab
        b_k1 = np.dot(A, b_k)
        # Calculate the norm
        b_k1_norm = np.linalg.norm(b_k1)
        # Re normalize the vector
        b_k = b_k1 / b_k1_norm
    return b_k

The power iteration converges to one of the eigenvectors associated with the largest eigenvalue, and iteratively updates and normalizes the vector at each step.

QR algorithms

The QR algorithm is a powerful method for solving the eigenvalue problem for a square matrix, especially when we need all the eigenvalues. This process usually works like this:

  1. Start with the matrix A 0 = A.
  2. Perform the QR decomposition: A k = Q k R k.
  3. Update A: A k+1 = R k Q k.
  4. Repeat until convergence.

Finally, A k is transformed into an upper triangular matrix, and the eigenvalues are the diagonal elements.

Example: calculating the eigenvalues of a matrix

To make things more clear, let's calculate the eigenvalues of a simple 2x2 matrix:

A = [4, 2]
[1, 3]

The characteristic equation is given as:

det(A - λI) = |4-λ 2 |
|1 3-λ| = 0

Expanding the determinant, we get:

(4-λ)(3-λ) - (2)(1) = λ 2 - 7λ + 10 = 0

Solving quadratic equations:

λ 1 = 5, λ 2 = 2

Thus, the eigenvalues of matrix A are 5 and 2.

Importance of eigenvectors

Once we find the eigenvalues, it becomes important to find the corresponding eigenvectors, as they give information about the direction of change.

For each eigenvalue, substitute v back into (A - λI)v = 0 to solve for λ. Taking λ = 5, we solve:

|4-5 2| |x| = |0|
|1 3-5| |y| = |0|

Leading up to the system:

-x + 2y = 0
x - 2y = 0

Solving gives the eigenvectors as any scalar multiple of [2, 1].

Applications in the real world

Engineering and vibration analysis

In engineering, eigenvalues are used mainly to analyze stability and vibration. For example, the natural frequencies of a mechanical structure are the eigenvalues of a system obtained from its mass and stiffness matrices.

Principal component analysis (PCA)

In statistics, PCA uses eigenvalue computation to reduce the dimensionality of a dataset, allowing for easier visualization and interpretation by finding “principal components.”

Data = [[2.5, 2.4], [0.5, 0.7], [2.2, 2.9], [1.9, 2.2], [3.1, 3.0], [2.3, 2.7], [2, 1.6], [1, 1.1], [1.5, 1.6], [1.1, 0.9]]
Covariance_matrix = np.cov(Data, rowvar=False)
eigen_values, eigen_vectors = np.linalg.eig(Covariance_matrix)

Here, the eigenvalues of the covariance matrix help to identify the main direction in which the dataset varies the most.

Challenges in eigenvalue computation

Although the concepts of eigenvalues and eigenvectors are simple, the calculations can be challenging, especially for large matrices:

  • Stability: Small changes in the matrix can lead to significant changes in the eigenvalues, especially for defective matrices.
  • Complexity: As the size of the matrix increases, the computation becomes intensive, requiring efficient algorithms and approximations.

Conclusion

Eigenvalue computations are essential in many computational algorithms used in both theoretical and applied science. A number of techniques ranging from basic power iterations to advanced QR algorithms are used to determine these values efficiently. Understanding the underlying principles facilitates their application in many fields, ensuring accurate solutions in scientific computing and beyond.


Graduate → 6.2.2


U
username
0%
completed in Graduate

← Prev (6.2.1)
Matrix Factorizations
Next (6.2.3) →
Sparse Matrices

Comments 



Eigenvalue Computations

https://www.buddymath.com/g/6.2.2?bmal=en