Number Representations & States

"how numbers are stored and used in computers"

Eigenvalues and Eigenvectors

An eigenvector is a non-zero vector that, when transformed by a linear transformation, only changes by a scalar factor (the eigenvalue).

Visualization Controls

Click on the canvas to select a vector

Grid shows unit vectors. Click anywhere to select a vector and see its transformation.

Green vectors are eigenvectors - they only scale, never rotate!

Matrix Controls

Transformation Matrix
2.000.000.003.00

Eigen Analysis

Eigenvectors are special vectors that don't change direction when transformed by a matrix.

Eigenvalues tell us how much the eigenvector gets scaled.

No real eigenvalues found (complex eigenvalues indicate rotation)

Selected Vector:

(1.000, 0.500)
Click on the canvas to select a different vector

Legend:

Original vectors
Transformed vectors
Eigenvectors

Definition

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

Then:

  • λ is called an eigenvalue of A
  • v is called an eigenvector of A corresponding to λ

Finding Eigenvalues and Eigenvectors

Characteristic Equation

To find eigenvalues, solve:

where I is the identity matrix of the same size as A.

Finding Eigenvectors

For each eigenvalue λ, solve:

Properties

  1. Eigenvalue Properties

    • The sum of eigenvalues equals the trace of A
    • The product of eigenvalues equals the determinant of A
    • Complex eigenvalues always come in conjugate pairs

  2. Eigenvector Properties

    • Eigenvectors corresponding to different eigenvalues are linearly independent
    • The number of linearly independent eigenvectors for an eigenvalue ≤ its algebraic multiplicity

Diagonalization

A matrix A is diagonalizable if:

where:

  • D is a diagonal matrix of eigenvalues
  • P is a matrix whose columns are eigenvectors

Applications

1. Principal Component Analysis (PCA)

  • Uses eigenvalues and eigenvectors for dimensionality reduction
  • Identifies principal directions of variation in data

2. Quantum Mechanics

  • Energy levels (eigenvalues) and states (eigenvectors)
  • Observable operators and measurement outcomes

3. Vibration Analysis

  • Natural frequencies (eigenvalues)
  • Mode shapes (eigenvectors)

4. Google's PageRank Algorithm

  • Web page importance as an eigenvector problem
  • Steady-state distribution of random walks

Computational Considerations

Time Complexity

  • Finding eigenvalues: O(n³) using QR algorithm
  • Finding eigenvectors: O(n³) per eigenvalue
  • Diagonalization: O(n³)

Space Complexity

  • O(n²) for storing matrices
  • O(n) for storing eigenvalues and individual eigenvectors

Numerical Methods

  1. Power Method

    • Finds dominant eigenvalue and eigenvector
    • Time complexity: O(kn²) for k iterations

  2. QR Algorithm

    • Finds all eigenvalues
    • More stable than direct characteristic equation solving

  3. Lanczos Algorithm

    • Efficient for sparse matrices
    • Finds extreme eigenvalues