Defective matrix

Inlinear algebra,adefective matrixis asquare matrixthat does not have a completebasisofeigenvectors,and is therefore notdiagonalizable.In particular, an$n\times n$matrixis defectiveif and only ifit does not have$n$linearly independenteigenvectors.A complete basis is formed by augmenting the eigenvectors withgeneralized eigenvectors,which are necessary for solving defective systems ofordinary differential equationsand other problems.

An$n\times n$defective matrix always has fewer than$n$distincteigenvalues,since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues$\lambda$withalgebraic multiplicity$m>1$(that is, they are multiplerootsof thecharacteristic polynomial), but fewer than$m$linearly independent eigenvectors associated with$\lambda$.If the algebraic multiplicity of$\lambda$exceeds itsgeometric multiplicity(that is, the number of linearly independent eigenvectors associated with$\lambda$), then$\lambda$is said to be adefective eigenvalue.However, every eigenvalue with algebraic multiplicity$m$always has$m$linearly independent generalized eigenvectors.

Arealsymmetric matrixand more generally aHermitian matrix,and aunitary matrix,is never defective; more generally, anormal matrix(which includes Hermitian and unitary matrices as special cases) is never defective.

Jordan block

Any nontrivialJordan blockof size$2\times 2$or larger (that is, not completely diagonal) is defective. (A diagonal matrix is a special case of the Jordan normal form with all trivial Jordan blocks of size$1\times 1$and is not defective.) For example, the$n\times n$Jordan block

$J={\begin{bmatrix}\lambda &1&\;&\;\\\;&\lambda &\ddots &\;\\\;&\;&\ddots &1\\\;&\;&\;&\lambda \end{bmatrix}},$

has aneigenvalue,$\lambda$with algebraic multiplicity$n$(or greater if there are other Jordan blocks with the same eigenvalue), but only one distinct eigenvector$Jv_{1}=\lambda v_{1}$,where$v_{1}={\begin{bmatrix}1\\0\\\vdots \\0\end{bmatrix}}.$The other canonical basis vectors$v_{2}={\begin{bmatrix}0\\1\\\vdots \\0\end{bmatrix}},~\ldots,~v_{n}={\begin{bmatrix}0\\0\\\vdots \\1\end{bmatrix}}$form a chain of generalized eigenvectors such that$Jv_{k}=\lambda v_{k}+v_{k-1}$for$k=2,\ldots,n$.

Any defective matrix has a nontrivialJordan normal form,which is as close as one can come todiagonalizationof such a matrix.

Example

A simple example of a defective matrix is

${\begin{bmatrix}3&1\\0&3\end{bmatrix}},$

which has a doubleeigenvalueof 3 but only one distinct eigenvector

${\begin{bmatrix}1\\0\end{bmatrix}}$

(and constant multiples thereof).

See also

• Jordan normal form– Form of a matrix indicating its eigenvalues and their algebraic multiplicities

References

• Golub, Gene H.; Van Loan, Charles F. (1996),Matrix Computations(3rd ed.), Baltimore:Johns Hopkins University Press,ISBN978-0-8018-5414-9
• Strang, Gilbert (1988).Linear Algebra and Its Applications(3rd ed.). San Diego: Harcourt.ISBN978-970-686-609-7.