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Introduction to Eigenvalues and Eigenvectors

What Are Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, appearing in many areas such as physics, computer science, and engineering.

Definition

  • Eigenvector: A non-zero vector \( \mathbf{v} \) such that when a linear transformation (matrix \( A \)) is applied to it, the vector is scaled by a scalar \( \lambda \):

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

  • Eigenvalue: The scalar \( \lambda \) that corresponds to an eigenvector \( \mathbf{v} \).

Key Properties

  1. Eigenvectors corresponding to distinct eigenvalues are linearly independent.
  2. If \( A \) is an \( n \times n \) matrix, there are at most \( n \) eigenvalues (some may be repeated).

Example

Given:

\[ A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \]

Find eigenvalues and eigenvectors.

Solution

  1. Compute \( \det(A - \lambda I) \):

    \[ \det\left(\begin{bmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \end{bmatrix}\right) = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 \]

    Solve \( \lambda^2 - 4\lambda + 3 = 0 \) to get \( \lambda_1 = 3 \), \( \lambda_2 = 1 \)

  2. Find eigenvectors for each eigenvalue:

    • For \( \lambda = 3 \): Solve \( (A - 3I) \mathbf{v} = 0 \).
    • For \( \lambda = 1 \): Solve \( (A - I) \mathbf{v} = 0 \).

The Characteristic Equation

Definition

The characteristic equation of a square matrix \( A \) is derived from \( \det(A - \lambda I) = 0 \). This polynomial equation determines the eigenvalues of \( A \).

Steps to Formulate

  1. Subtract \( \lambda I \) from \( A \):

    \[ A - \lambda I \]

  2. Compute the determinant:

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

  3. Set \( \det(A - \lambda I) = 0 \).

Example

For \( A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix} \):

  1. \( A - \lambda I = \begin{bmatrix} 4-\lambda & 2 \\ 1 & 3-\lambda \end{bmatrix} \).
  2. \( \det(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 \cdot 1 = \lambda^2 - 7\lambda + 10 \).
  3. Solve \( \lambda^2 - 7\lambda + 10 = 0 \) to get \( \lambda = 5, 2 \).

Diagonalization

What Is Diagonalization?

A matrix \( A \) is diagonalizable if it can be expressed as:

\[ A = PDP^{-1} \]

where:

  • \( P \) is a matrix whose columns are the eigenvectors of \( A \).
  • \( D \) is a diagonal matrix with eigenvalues of \( A \) on its diagonal.

Conditions for Diagonalization

  1. \( A \) must have \( n \) linearly independent eigenvectors.
  2. \( A \) must be a square matrix.

Steps to Diagonalize

  1. Find eigenvalues \( \lambda_1, \lambda_2, \dots \).
  2. Find eigenvectors for each eigenvalue.
  3. Form \( P \) using eigenvectors as columns.
  4. Form \( D \) with eigenvalues along the diagonal.
  5. Compute \( P^{-1} \).

Example

For \( A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \):

  1. Eigenvalues: \( \lambda_1 = 5, \lambda_2 = 2 \).
  2. Eigenvectors: \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \).
  3. \( P = \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} \), \( D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix} \).
  4. \( P^{-1} = \begin{bmatrix} 1/3 & 1/3 \\ -2/3 & 1/3 \end{bmatrix} \).
  5. Verify: \( A = PDP^{-1} \).

Eigenvectors and Linear Transformations

Geometric Interpretation

An eigenvector represents a direction that remains unchanged under the linear transformation defined by \( A \), except for scaling by the eigenvalue \( \lambda \).

Applications

  • Principal Component Analysis (PCA): Eigenvectors represent principal directions of data variance.
  • Quantum Mechanics: Eigenvalues correspond to measurable quantities.
  • Graph Theory: Eigenvectors indicate centrality in networks.

Example

For a transformation \( A \):

\[ A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x + y \\ x + 2y \end{bmatrix} \]

Find eigenvalues and interpret geometrically.

Complex Eigenvalues

Definition

If \( A \) has complex eigenvalues, they appear in conjugate pairs \( \lambda = a \pm bi \).

Example

For \( A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \): 1. \( \det(A - \lambda I) = \lambda^2 + 1 = 0 \). 2. Eigenvalues: \( \lambda = i, -i \). 3. Eigenvectors involve complex numbers, e.g., \( \mathbf{v} = \begin{bmatrix} i \\ 1 \end{bmatrix} \).

Discrete Dynamical Systems

Definition

A discrete dynamical system evolves in discrete time steps according to:

\[ \mathbf{x}_{n+1} = A \mathbf{x}_n \]

where \( A \) is the transition matrix.

Stability Analysis

  1. Compute eigenvalues of \( A \).
  2. If all \( |\lambda| < 1 \), the system is stable.

Example

For \( A = \begin{bmatrix} 0.5 & 0.5 \\ 0.2 & 0.8 \end{bmatrix} \): 1. Eigenvalues: \( \lambda = 1, 0.3 \). 2. System is stable as \( |\lambda| < 1 \).

Applications

  • Population models
  • Economic systems
  • Markov chains