Eigenvectors and Eigenvalues

Eigenvectors and Eigenvalues

Definition and Concepts

  • Eigenvectors are special vectors associated with a matrix, which, when multiplied by a matrix, yield a scaled version of the original vector.
  • The scalar factor is known as the eigenvalue corresponding to that eigenvector.
  • More formally, if a matrix A and a vector v satisfy the equation Av = λv, then v is an eigenvector of A and λ is the corresponding eigenvalue.

Calculating Eigenvectors and Eigenvalues

  • The first step is to determine the eigenvalues of a matrix. This is done by solving the characteristic equation, ** A - λI = 0**, where ‘I’ is an identity matrix.
  • After the eigenvalues have been found, eigenvectors can be calculated by solving the system of linear equations formed by (A - λI)v = 0.
  • It’s important to remember that eigenvectors are only determined up to an arbitrary scalar. If v is an eigenvector of A, then so is any scalar multiple of v.

Properties of Eigenvectors and Eigenvalues

  • Eigenvectors corresponding to distinct eigenvalues are linearly independent.
  • The determinant of a matrix is equal to the product of its eigenvalues.
  • The trace (the sum of the entries on the main diagonal) of a matrix is equal to the sum of its eigenvalues.
  • Eigenvectors and their respective eigenvalues provide insightful information about the matrix, including its determinant, trace and linear transformations it represents.

Applications of Eigenvectors and Eigenvalues

  • In differential equations, eigenvalues and eigenvectors are used to find solutions to systems of differential equations.
  • They are also used in the diagonalisation of matrices, a process that simplifies many matrix operations.
  • In physics, they are used in problems dealing with normal modes of oscillation.
  • In data analysis, eigenvalues and eigenvectors play an essential role in Principal Component Analysis (PCA), a popular technique for dimensionality reduction.
  • In network theory, Google’s search algorithm (PageRank) uses concepts of eigenvalues and eigenvectors.

Complex Eigenvalues and Eigenvectors

  • Eigenvectors and eigenvalues can be complex numbers.
  • When a matrix has complex eigenvalues, it corresponds to a rotational (as opposed to stretching/squishing) linear transformation.
  • Complex eigenvalues and eigenvectors follow the same rules as their real-number counterparts.