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| Diagonalization is useful primarily for repeated multiplication by a matrix. '''''A'''^2^ = '''SΛ'''^2^'''S'''^-1^'', and more generally '''''A'''^K^ = '''SΛ'''^K^'''S'''^-1^''. | Diagonalization offers clean solutions to mathematical models. '''''A'''^2^ = '''SΛ'''^2^'''S'''^-1^'', and more generally '''''A'''^K^ = '''SΛ'''^K^'''S'''^-1^''. Similarly, ''e'''^A^''' = '''S'''e'''^Λ^S'''^-1^''. Note that ''e'''^Λ^''''' is a diagonal matrix with ''e'' to the power of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] of '''''A''''' in the diagonal. |
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.