Diagonalization
Diagonalization is an alternative decomposition of square matrices.
Contents
Introduction
A diagonal matrix has many useful properties. A diagonalizable matrix is a square matrix that can be factored into one. A square matrix that cannot be factored like this is called defective.
Definition
Given a matrix A, notate the matrix of its eigenvectors as S. A diagonalizable matrix can be factored as A = SΛS-1.
Λ will be a diagonal matrix with the eigenvalues of A in the diagonal.
In other words, A can be rewritten as a eigennormalized (i.e. transformed by S) then un-eigennormalized (i.e. transformed by S-1) diagonal matrix Λ.
Properties
Diagonalization offers clean solutions to mathematical models.
A2 = SΛ2S-1, and more generally AK = SΛKS-1.
Similarly, eA = SeΛS-1. Note that eΛ is a diagonal matrix with e to the power of the eigenvalues of A in the diagonal.