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In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not 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 eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems.

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rdf:type	anatomical structure
rdfs:label	Defective matrix (en)
rdfs:comment	In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not 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 eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. (en)
sameAs	Defective matrix
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_book dbt:Reflist dbt:Short_description dbt:Use_American_English dbt:Matrix_classes
Subject	Linear algebra
Link from a Wikipage to an external page	https://archive.org/details/linearalgebraits00stra%7Curl-access=registration%7Cedition=3rd
gold:hypernym	Matrix
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Defective_matrix?oldid=1070866177&ns=0
Wikipage page ID	5302952 (xsd:integer)
page length (characters) of wiki page	4067 (xsd:nonNegativeInteger)
Wikipage revision ID	1070866177 (xsd:integer)
Link from a Wikipage to another Wikipage	Normal matrix Unitary matrix Basis (linear algebra) Generalized eigenvector Jordan matrix Jordan normal form Linear algebra Zero of a function Characteristic polynomial Linear algebra Linear independence Square matrix Eigenvalues and eigenvectors Hermitian matrix Ordinary differential equation If and only if Symmetric matrix Diagonalizable matrix Johns Hopkins University Press Matrix (mathematics) Real number
has abstract	In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not 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 eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. An n × n defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with λ. If the algebraic multiplicity of λ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with λ), then λ is said to be a defective eigenvalue. However, every eigenvalue with algebraic 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. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Defective_matrix
is Wikipage redirect of	Defective Matrix Defective eigenvalue Defective matrices
is Link from a Wikipage to another Wikipage of	Generalized eigenvector Jordan normal form Discrete Fourier transform Defective Defective Matrix List of named matrices Eigendecomposition of a matrix Eigenvalues and eigenvectors Diagonalizable matrix Involutory matrix Markov chain Expm Defective eigenvalue Defective matrices
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Defective_matrix

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