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

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