Complete Guide to Eigenvalues, Eigenvectors & Matrix Diagonalization
Eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications throughout mathematics, physics, engineering, and data science. For a square matrix $A$, an eigenvector $v$ satisfies $Av = \lambda v$, where $\lambda$ is the corresponding eigenvalue. This relationship reveals the matrix's intrinsic geometric properties.
🔢 What are Eigenvalues?
Eigenvalues are scalars that represent how much an eigenvector is stretched or compressed during the linear transformation. They solve the characteristic equation $\det(A - \lambda I) = 0$.
📐 Geometric Interpretation
Eigenvectors point in directions that remain unchanged (except scaling) under the transformation. Eigenvalues determine the scaling factor in those invariant directions.
⚙️ Physical Applications
In quantum mechanics, eigenvalues represent measurable quantities. In vibration analysis, they determine natural frequencies. In statistics, they appear in principal component analysis.
The Characteristic Polynomial Method
The standard approach to finding eigenvalues involves computing the characteristic polynomial:
For an $n \times n$ matrix, this yields an $n$th-degree polynomial whose roots are the eigenvalues. Our calculator shows the complete step-by-step determinant calculation.
Finding Eigenvectors: Solving (A - λI)v = 0
Once eigenvalues $\lambda_i$ are found, eigenvectors are obtained by solving the homogeneous system:
The solution space is called the eigenspace corresponding to $\lambda_i$. Its dimension is the geometric multiplicity of the eigenvalue.
Matrix Diagonalization: A = PDP⁻¹
A matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and diagonal matrix $D$ such that:
Where $D$ contains eigenvalues on the diagonal, and $P$ contains corresponding eigenvectors as columns. Diagonalization simplifies matrix powers and exponentials.
Spectral Theorem and Symmetric Matrices
For real symmetric matrices (or complex Hermitian matrices), the Spectral Theorem guarantees:
✅ Real Eigenvalues
All eigenvalues are real numbers, never complex (for symmetric/Hermitian matrices).
✅ Orthogonal Eigenvectors
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
✅ Full Diagonalization
Symmetric matrices are always diagonalizable with an orthogonal matrix $P$.
Algebraic vs. Geometric Multiplicity
Understanding these two multiplicities is crucial for diagonalizability:
Algebraic Multiplicity
The number of times $\lambda$ appears as a root of the characteristic polynomial. If $(\lambda - \lambda_i)^m$ is a factor, algebraic multiplicity = $m$.
Geometric Multiplicity
The dimension of the eigenspace $E_{\lambda_i} = \ker(A - \lambda_i I)$. Equals the number of linearly independent eigenvectors for $\lambda_i$.
Key Fact: A matrix is diagonalizable if and only if for each eigenvalue, algebraic multiplicity = geometric multiplicity.
Applications in Quantum Mechanics and PCA
Eigenvalue problems appear throughout advanced mathematics and applications:
⚛️ Quantum Mechanics
Observables correspond to Hermitian operators. Measurement outcomes are eigenvalues, and state collapse yields eigenvectors.
📊 Principal Component Analysis
PCA finds eigenvectors of the covariance matrix (principal components) with largest eigenvalues explaining maximum variance.
🌊 Vibration Analysis
Natural frequencies of mechanical systems are square roots of eigenvalues of the stiffness matrix.
🚀 Master Spectral Theory
Our eigenvalue calculator is part of a comprehensive linear algebra toolkit. Explore related topics and advanced theory.
Computational Methods for Large Matrices
While our calculator handles matrices up to 5×5 with exact methods, industrial applications require specialized algorithms:
Power Iteration Method
Finds the dominant eigenvalue (largest magnitude) and corresponding eigenvector through repeated multiplication: $v_{k+1} = Av_k / \|Av_k\|$.
QR Algorithm
Industry standard for finding all eigenvalues of dense matrices. Based on QR decomposition and similarity transformations.
Lanczos Algorithm
For large sparse symmetric matrices. Constructs tridiagonal matrix with same eigenvalues, then applies efficient methods.
Common Eigenvalue Problems and Solutions
| Matrix Type | Eigenvalue Properties | Diagonalizability |
|---|---|---|
| Symmetric (Real) | All eigenvalues real, orthogonal eigenvectors | Always diagonalizable |
| Diagonal Matrix | Diagonal entries are eigenvalues | Already diagonal |
| Triangular Matrix | Diagonal entries are eigenvalues | May not be diagonalizable |
| Projection Matrix | Eigenvalues 0 or 1 only | Always diagonalizable |
Eigenvalue Calculator Algorithm and Implementation
Our calculator implements exact mathematical methods for matrices up to 5×5:
- Input Validation: Verify matrix is square and contains valid numbers
- Characteristic Polynomial: Compute $\det(A - \lambda I)$ symbolically
- Root Finding: Solve polynomial equation for eigenvalues (exact for ≤4, numerical for 5)
- Eigenspace Computation: For each eigenvalue, solve $(A - \lambda I)v = 0$
- Diagonalization Check: Verify algebraic = geometric multiplicities
- Result Formatting: Present eigenvalues, eigenvectors, and diagonalization
❓ Eigenvalue Calculator FAQ
What if my matrix has complex eigenvalues?
Our calculator handles complex eigenvalues and eigenvectors. For real matrices, complex eigenvalues occur in conjugate pairs with corresponding complex eigenvectors.
How do I know if a matrix is diagonalizable?
A matrix is diagonalizable if for each eigenvalue, algebraic multiplicity equals geometric multiplicity. Our calculator checks this condition and reports diagonalizability.
What is the difference between eigenvalues and singular values?
Eigenvalues apply to square matrices and solve $Av = \lambda v$. Singular values apply to any matrix and are square roots of eigenvalues of $A^TA$. Both reveal matrix structure.
Can I compute eigenvalues of non-square matrices?
Eigenvalues are defined only for square matrices. For non-square matrices, compute singular values instead (see our future SVD calculator).