= Diagonalization =

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

<<TableOfContents>>

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== Description ==

A '''diagonal matrix''' is a diagonal line of numbers in a square matrix of zeros.

Such a matrix has many useful properties.
 * Its columns are its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]]
 * The numbers in the diagonal are the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]
 * The [[LinearAlgebra/Determinant|determinant]] is the project of the numbers in the diagonal

'''Diagonalization''' is a decomposition that turns a matrix into a diagonal matrix, with some change of basis matrices to the left and right of it. The idea follows from eigenvectors:
 * Recall that '''''A'''x = λx''.
 * It follows that '''''AS''' = '''SΛ''''' where...
   * '''''S''''' is the eigenbasis composed of all eigenvectors ''x''.
   * '''''Λ''''' is the diagonal matrix where the numbers in the diagonal are the eigenvalues.

Recall that only square matrices have eigenvectors, and that a matrix of size ''n x n'' either has ''n'' unique eigenpairs or is '''defective'''. These restrictions apply to diagonalization; a matrix that cannot be diagonalized is defective.

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== Procedure ==

The procedure for diagonalizing '''''A''''' is:
 * solving for [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues and eigenvectors]]
 * constructing '''''S''''' and '''''Λ''''' as described above
   * Note that order of eigenpairs does not matter, as long as the orders in '''''S''''' and '''''Λ''''' match.
 * [[LinearAlgebra/Invertibility|inverting]] '''''S'''''

In the case of a [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric]] matrix '''''A''''', the eigenbasis can be [[LinearAlgebra/Transposition|transposed]] instead of calculating the inverse. To indicate this, the eigenbasis is generally notated as '''''Q''''' instead, as in '''''A''' = '''QΛQ'''^T^''.

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== Usage ==

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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