Eigenvalue and Eigenvector

Eigenvalue and Eigenvector

• Eigenvalue and Eigenvector are concepts that represent core characteristics of a matrix in linear algebra
• For square matrix A, eigenvector is a non-zero vector that direction is unchanged when linear transformation is applied and agenvalue represents the rate of change at this time. Mathmatically represented as $Ax = {\lambda}x$.

Meaning of Eigenvalue and Eigenvector

Attribute of linear transformation

• Matrix A executes linear transformation (roation, expansion, reduction, etc…) in a vector space. The directon of eigenvector is identical after the transformation and eigenvalue ${\lambda}$ means scaling the vector

  ${\lambda}$

  times.

  • ${\lambda}>1$: vector expansion
  • $0 < {\lambda} <1$: vector reduction
  • ${\lambda}<0$: direction reversal

Mathmatical definition

• Eigenvalue is calculated by solving characteristics equation $det(A - {\lambda}I) = 0$ and eigenvector is decided as the value of $(A - {\lambda}I)x = 0$

Importance and Application Domain

Dimension Reduction and Data Analysis

PCA (Principal Component Analysis)

• Eigenvector of a covariance matrix represents maximum variance directon of data and eigenvalue decides importance of the direction. Through this, high-demensional data can be reduced into low-dimensional data.

  System Analysis

  Vibration Analysis

• Evaluate stability of a structure by calculating natural frequency(eigenvalue) and frequency mode(eigenvector) of a machine system.

  Quantum Dynamics

• Physical quantity like energy level is represented as eigenvalue of an operator.

  Calculation Efficiency

• Complexity is dramatically reduced when a matrix’s involution $A^k$ is calculated with eigensolution. For example, when decomposed as $A = PDP^{-1}$, $A^k = PD^kP^{-1}$ is validated.

  Graph Theory

• Eigenvalue of an adjacency matrix is used for analyzing network connectivity, community structure and etc.

Example

• For matrix $A = {\begin{pmatrix}2 & 1 \ 1 & 2\end{pmatrix}}$,
  • Eigenvalue: ${\lambda_1 = 3, \lambda_2 = 1}$
  • Eigenvector: $v_1 = \begin{pmatrix}1 \ 1\end{pmatrix}, v_2 = \begin{pmatrix}-1 \ 1\end{pmatrix}$