en>Mogism: /* History */Cleanup/Typo fixing, replaced: was build → was built using AWB

2014-01-24T23:48:31Z

History: Cleanup/Typo fixing, replaced: was build → was built using AWB

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In [[linear algebra]], a '''defective matrix''' is a [[square matrix]] that does not have a complete [[basis function|basis]] of [[eigenvector]]s, and is therefore not [[diagonalizable matrix|diagonalizable]]. In particular, an ''n'' × ''n'' matrix is defective if and only if it does not have ''n'' [[linearly independent]] eigenvectors. A complete basis is formed by augmenting the eigenvectors with [[generalized eigenvector]]s, which are necessary for solving defective systems of [[ordinary differential equation]]s and other problems.

A defective matrix always has fewer than ''n'' distinct [[eigenvalue]]s, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with [[algebraic multiplicity]] <math>m > 1</math> (that is, they are multiple roots of the [[characteristic polynomial]]), but fewer than ''m'' linearly independent eigenvectors associated with λ. However, every eigenvalue with multiplicity ''m'' always has ''m'' linearly independent generalized eigenvectors.

A [[Hermitian matrix]] (or the special case of a real [[symmetric matrix]]) or a [[unitary matrix]] is never defective; more generally, a [[normal matrix]] (which includes Hermitian and unitary as special cases) is never defective.

== Jordan block ==
Any [[Jordan normal form|Jordan block]] of size 2×2 or larger is defective. For example, the n × n Jordan block,
:<math>J =
\begin{bmatrix}
\lambda & 1 & \; & \; \\
\; & \lambda & \ddots & \; \\
\; & \; & \ddots & 1 \\
\; & \; & \; & \lambda
\end{bmatrix},</math>
has an [[eigenvalue]], λ, with multiplicity n, but only one distinct eigenvector,
:<math>v = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}.</math>

== Example ==

A simple example of a defective matrix is:
:<math>\begin{bmatrix} 3& 1 \\ 0 & 3 \end{bmatrix}</math>
which has a double [[eigenvalue]] of 3 but only one distinct eigenvector
:<math>\begin{bmatrix} 1 \\ 0 \end{bmatrix}</math>
(and constant multiples thereof).

== See also ==
* [[Jordan normal form]]

==References==
* {{cite book|first=Gilbert|last= Strang|title=Linear Algebra and Its Applications|edition=3rd |publisher=Harcourt|location= San Diego|year= 1988|isbn=970-686-609-4}}

[[Category:Linear algebra]]

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en>Mogism: /* History */Cleanup/Typo fixing, replaced: was build → was built using AWB