= Diagonalization =

'''Diagonalization''' is an alternative decomposition of square matrices.

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== Introduction ==

A [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] has many useful properties. A [[LinearAlgebra/MatrixProperties#Diagonalizability|diagonalizable matrix]] is a ''square'' matrix that can be factored into one. A square matrix that cannot be factored like this is called '''defective'''.

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== Definition ==

Given a matrix '''''A''''', notate the matrix of its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] as '''''S'''''. A diagonalizable matrix can be factored as '''''A''' = '''SΛS'''^-1^''.

'''''Λ''''' will be a diagonal matrix with the [[LinearAlgebra/EigenvaluesAndEigenvectors|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.

'''''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.



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