(Redirected page to Eigendecomposition)
 
 
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#REDIRECT [[Eigendecomposition]]
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== Diagonalization ==
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Diagonalization is the process of tranforming a square matrix $A$ to the diagonal form
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Diagonal Form
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* $T$ is the ''diagonal form'' of $A$ if
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* $T$ is Diagonal and
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* there exists $X$ such that $T = X^{-1} A X$
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* $A$ and $T$ are [[Similar Matrices|similar]]
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* so $A$ and $T$ share the same [[Eigenvalues and Eigenvectors|eigenvalues]]
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Non-defectiveness
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* $A$ is ''non defective'' $\iff$ there exists non singular $X$ s.t.
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* $X^{-1} A X = \text{diag}(\lambda_1, ..., \lambda_n)$
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* i.e. there exists a similarity tranformation $X$ such that the results is a diagonal matrix with eigenvalues on the diagonal
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== [[Eigendecomposition]] ==
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Eigendecomposition decomposes a symmetric matric $A$ as
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* $\Lambda = S^{T} A S$
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* where $\Lambda = \text{diag}(\lambda_1, ..., \lambda_n)$
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* and $S$ has eigenvectors on the diagonal
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* so the similarity transformation matrix $S$ here is [[Orthogonal Matrices|orthogonal]]
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== Sources ==
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* [[Linear Algebra MIT 18.06 (OCW)]]
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* [[Matrix Computations (book)]]
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[[Category:Linear Algebra]]

Latest revision as of 16:55, 28 June 2017

Diagonalization

Diagonalization is the process of tranforming a square matrix $A$ to the diagonal form

Diagonal Form

  • $T$ is the diagonal form of $A$ if
  • $T$ is Diagonal and
  • there exists $X$ such that $T = X^{-1} A X$
  • $A$ and $T$ are similar
  • so $A$ and $T$ share the same eigenvalues


Non-defectiveness

  • $A$ is non defective $\iff$ there exists non singular $X$ s.t.
  • $X^{-1} A X = \text{diag}(\lambda_1, ..., \lambda_n)$
  • i.e. there exists a similarity tranformation $X$ such that the results is a diagonal matrix with eigenvalues on the diagonal


Eigendecomposition

Eigendecomposition decomposes a symmetric matric $A$ as

  • $\Lambda = S^{T} A S$
  • where $\Lambda = \text{diag}(\lambda_1, ..., \lambda_n)$
  • and $S$ has eigenvectors on the diagonal
  • so the similarity transformation matrix $S$ here is orthogonal


Sources