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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 '''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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| == Definition == | == Procedure == |
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| Given a matrix '''''A''''', notate the matrix of its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] as '''''S'''''. A diagonalizable matrix can be factored as '''''A''' = '''SΛS'''^-1^''. | 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''''' |
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| '''''Λ''''' will be a diagonal matrix with the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] of '''''A''''' in the diagonal. | 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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| 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'''''. | ---- |
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| === Properties === | == Usage == |
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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
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
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 Ax = 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.
Procedure
The procedure for diagonalizing A is:
solving for 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.
In the case of a symmetric matrix A, the eigenbasis can be transposed instead of calculating the inverse. To indicate this, the eigenbasis is generally notated as Q instead, as in A = QΛQT.
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.