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| == Introduction == | == Description == |
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| 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'''. | 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 A matrix is '''diagonalizable''' if it can be factored into a diagonal matrix. Only square matrices can be diagonalizable. A square matrix that still cannot be factored as such is '''defective'''. |
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| == Definition == | == Process == |
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| === Properties === | |
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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^''. | == 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. |
Diagonalization
Diagonalization is an alternative decomposition of square matrices.
Contents
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 eigenvectors
The numbers in the diagonal are the eigenvalues
The determinant is the project of the numbers in the diagonal
A matrix is diagonalizable if it can be factored into a diagonal matrix. Only square matrices can be diagonalizable. A square matrix that still cannot be factored as such is defective.
Process
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.
Usage
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.