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Linear Algebra, Optimisation and Numerical Methods / Eigenvalues, Diagonalisation, and Dynamical Applications

Diagonalisation: A = PDP⁻¹

Example 2 of the change-of-basis note diagonalised a matrix by a "lucky guess" of a good basis. This note reveals the guess was never luck: the basis that diagonalises AA is always a basis of eigenvectors, when one exists. It gives the general formula A=PDP−1A=PDP^{-1}, shows why this makes computing AkA^k almost free, and states precisely — via algebraic and geometric multiplicity — when a full eigenvector basis fails to exist.
Learning objectives. By the end of this note you should be able to:
  • assemble the diagonalisation A=PDP−1A=PDP^{-1} from a full set of eigenpairs;
  • use diagonalisation to compute AkA^k efficiently;
  • define algebraic and geometric multiplicity and state the diagonalisability criterion in terms of them;
  • recognise when a matrix is not diagonalisable from its eigenstructure.

In this lesson

  1. The diagonalisation formula
  2. Powers of AA, almost free
  3. Algebraic and geometric multiplicity

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