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MATH
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Definitions from Diagonalisation

All definitions from Diagonalisation.

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Eigenbasis

An eigenbasis for a vector space is a basis whose vectors are eigenvectors of the linear map under consideration.

In an eigenbasis, each coordinate direction is scaled independently. Every vector therefore splits into eigenmodes, which matrix powers and linear ODE systems evolve separately.

  1. [1]:Eigenvector
  2. [2]:Basis
  3. [3]:Diagonalisation
  4. [4]:When diagonalisation is possible
  5. [5]:Real symmetric matrices

Diagonalisation

An n×nn\times n matrix AA over a field FF is diagonalisable over FF if there are an invertible matrix P∈Mn(F)P\in M_n(F) and a diagonal matrix D∈Mn(F)D\in M_n(F) such that

P−1AP=D,equivalentlyA=PDP−1.P^{-1}AP=D, \qquad\text{equivalently}\qquad A=PDP^{-1}.

The columns of PP are the eigenvectors in the chosen eigenbasis and the corresponding eigenvalues appear on the diagonal of DD.

  1. [1]:Eigenbasis
  2. [2]:Matrix representation of a linear map
  3. [3]:When diagonalisation is possible
  4. [4]:A real matrix with complex eigenvalues
  5. [5]:A repeated eigenvalue without enough modes
  6. [6]:A triangular matrix with an eigenbasis
  7. 13 more

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EigenbasisDiagonalisation