Editing Jordan normal form (section)

== Numerical analysis ==

If the matrix ''A'' has multiple eigenvalues, or is close to a matrix with multiple eigenvalues, then its Jordan normal form is very sensitive to perturbations. Consider for instance the matrix
:<math> A = \begin{bmatrix} 1 & 1 \\ \varepsilon & 1 \end{bmatrix}. </math>
If ''ε'' = 0, then the Jordan normal form is simply
:<math> \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}. </math>
However, for ''ε'' ≠ 0, the Jordan normal form is
:<math> \begin{bmatrix} 1+\sqrt\varepsilon & 0 \\ 0 & 1-\sqrt\varepsilon \end{bmatrix}. </math>
This [[condition number|ill conditioning]] makes it very hard to develop a robust numerical algorithm for the Jordan normal form, as the result depends critically on whether two eigenvalues are deemed to be equal. For this reason, the Jordan normal form is usually avoided in [[numerical analysis]]; the stable [[Schur decomposition]]<ref>See Golub & Van Loan (2014), §7.6.5; or Golub & Wilkinson (1976) for details.</ref> or [[pseudospectrum|pseudospectra]]<ref>See Golub & Van Loan (2014), §7.9</ref> are better alternatives.