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Difference between revisions of "Defective matrix"

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A [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110100/d1101001.png" /> is called non-defective if it has a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110100/d1101002.png" /> independent eigenvectors (cf. [[Eigen vector|Eigen vector]]). Otherwise it is called defective. The notion is of particular importance in numerical [[Linear-algebra(2)|linear algebra]].
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A [[matrix]] $A\in\mathbf C^{n\times n}$ is called non-defective if it has a set of $n$ independent eigenvectors (cf. [[Eigen vector]]). Otherwise it is called defective. The notion is of particular importance in numerical [[Linear-algebra(2)|linear algebra]].
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====Comment====
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A complex matrix $A$ is non-defective if and only if it is [[Similar matrices|similar]] to a [[diagonal matrix]]: $\Delta = P A P^{-1}$.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.M. Young,  R.T. Gregory,   "A survey of numerical mathematics" , '''2''' , Dover, reprint  (1988)  pp. 741–743</TD></TR></table>
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* D.M. Young,  R.T. Gregory, "A survey of numerical mathematics" , '''2''' , Dover, reprint  (1988)  pp. 741–743
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* Lloyd N. Trefethen, David Bau III, ''Numerical Linear Algebra'' SIAM (1997) {{ISBN|0898713617}}
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[[Category:Special matrices]]
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[[Category:Numerical analysis and scientific computing]]

Latest revision as of 14:06, 19 November 2023

A matrix $A\in\mathbf C^{n\times n}$ is called non-defective if it has a set of $n$ independent eigenvectors (cf. Eigen vector). Otherwise it is called defective. The notion is of particular importance in numerical linear algebra.

Comment

A complex matrix $A$ is non-defective if and only if it is similar to a diagonal matrix: $\Delta = P A P^{-1}$.

References

  • D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , 2 , Dover, reprint (1988) pp. 741–743
  • Lloyd N. Trefethen, David Bau III, Numerical Linear Algebra SIAM (1997) ISBN 0898713617
How to Cite This Entry:
Defective matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defective_matrix&oldid=13784
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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