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Every invertible matrix can be written as:
where:
Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.
Intuitively, Note that this is just the change of bases formula, and so:
  • changes basis to align to the eigenvectors
  • multiplies eigenvectors simply by eigenvalues
  • changes back to the original basis

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