Eigenvalue

An eigenvalue is a fundamental concept in linear algebra that emerges when studying how linear transformation affect vectors in space. When a linear transformation is applied to certain special vectors (called eigenvector), the result is simply a scaling of the original vector by a factor λ (lambda) - this scaling factor is the eigenvalue.

Mathematical Definition

For a square matrix A and a non-zero vector v, if there exists a scalar λ such that:

Av = λv

Then λ is an eigenvalue of A, and v is its corresponding eigenvector.

Properties

1. Characteristic Equation

   • Found by solving det(A - λI) = 0
   • The degree of this equation equals the matrix dimension
   • Complex eigenvalues can occur in conjugate pairs

2. Key Characteristics

   • The sum of eigenvalues equals the matrix trace
   • The product of eigenvalues equals the determinant
   • symmetric matrix have only real eigenvalues

Applications

Physics and Engineering

• quantum mechanics uses eigenvalues to represent observable quantities
• vibration analysis studies employ eigenvalues to find natural frequencies
• principal component analysis relies on eigenvalues for dimensionality reduction

Computer Science

• PageRank uses eigenvalues to rank web pages
• machine learning algorithms utilize eigenvalues for feature extraction
• image processing techniques often involve eigenvalue decomposition

Geometric Interpretation

Eigenvalues represent how vectors are stretched or compressed by a linear transformation. This leads to three possible scenarios:

1. λ > 1: The eigenvector is stretched
2. 0 < λ < 1: The eigenvector is compressed
3. λ < 0: The eigenvector is reversed and scaled

Historical Context

The concept emerged from the work of Euler in the 18th century, though the term "eigen" (meaning "characteristic" or "proper" in German) was introduced by David Hilbert in the early 20th century.

Computational Methods

Several algorithms exist for finding eigenvalues:

• power method
• QR algorithm
• Lanczos algorithm

These methods are particularly important for large matrices where direct calculation becomes impractical.

See Also

• eigendecomposition
• singular value
• characteristic polynomial
• Jordan canonical form