Eigen Library

The eigen library provides tools for eigenvalue problems, matrix diagonalization, spectral decomposition, and advanced matrix analysis. These concepts are fundamental to quantum mechanics, vibration analysis, principal component analysis (PCA), and many other applications in science and engineering.

What is Eigen Analysis?

Eigenvalues and eigenvectors reveal the intrinsic properties of linear transformations. When a matrix acts on an eigenvector, it only scales it by the corresponding eigenvalue, making these concepts essential for understanding matrix behavior, stability analysis, and dimensionality reduction.

PCA Vibration Analysis Quantum Mechanics Stability Analysis PageRank

Eigenvalue & Eigenvector Solvers

Core

Eigenvalue/Eigenvector Solver

Compute eigenvalues and eigenvectors with algebraic & geometric multiplicities. Get characteristic polynomial and spectral decomposition.

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Polynomial

Characteristic Polynomial

Find det(A - λI) expanded to polynomial form. Essential for finding eigenvalues and understanding matrix properties.

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Decomposition

Matrix Diagonalization

Find P and D such that A = PDP⁻¹. Check diagonalizability and compute matrix powers efficiently.

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Matrix Property Checkers

Checker

Symmetric Matrix Checker

Verify if a matrix is symmetric (A = Aᵀ). Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

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Checker

Positive Definite Checker

Test if all eigenvalues > 0, leading principal minors > 0, and xᵀAx > 0 for all non-zero x. Essential for optimization and Cholesky decomposition.

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Checker

Positive Semidefinite Checker

Test if all eigenvalues ≥ 0 and xᵀAx ≥ 0 for all x. Important for covariance matrices and quadratic forms.

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Advanced Eigen Analysis

Decomposition

Singular Value Decomposition (SVD)

Decompose any matrix into A = UΣVᵀ. Find singular values, left/right singular vectors. Fundamental for PCA, data compression, and pseudoinverses.

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Matrix Functions & Differential Equations

ODE

Linear ODE System Solver

Solve dx/dt = Ax with given initial conditions using eigenvalues/matrix exponential. Get general and particular solutions step-by-step.

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Understanding Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that reveal the intrinsic properties of linear transformations. When a matrix A acts on one of its eigenvectors v, the result is simply the same vector scaled by the corresponding eigenvalue λ:

\[ A\vec{v} = \lambda\vec{v} \]

Why Are Eigenvalues Important?

Principal Component Analysis (PCA): The eigenvectors of a covariance matrix identify the directions of maximum variance in high-dimensional data.
Quantum Mechanics: Observables are represented by Hermitian operators whose eigenvalues are the possible measurement outcomes.
Vibration Analysis: The natural frequencies of a mechanical system are the eigenvalues of its stiffness matrix divided by mass.
Stability Analysis: The stability of dynamical systems is determined by the eigenvalues of the system matrix (real parts must be negative).
Google PageRank: The PageRank algorithm finds the dominant eigenvector of the web's link matrix.

How to Find Eigenvalues

Eigenvalues are the roots of the characteristic polynomial:

\[ \det(A - \lambda I) = 0 \]

For each eigenvalue λ, we find the corresponding eigenvectors by solving:

\[ (A - \lambda I)\vec{v} = \vec{0} \]

Multiplicity Considerations

Algebraic Multiplicity: The number of times an eigenvalue appears as a root of the characteristic polynomial.
Geometric Multiplicity: The dimension of the eigenspace, i.e., the number of linearly independent eigenvectors associated with the eigenvalue.
A matrix is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity for every eigenvalue.

💡 Quick Fact:

The sum of all eigenvalues equals the trace of the matrix, and the product of all eigenvalues equals the determinant.