# Defective matrix

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.[1] 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.

A 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**.[1] 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.

## Jordan block

Any nontrivial Jordan block of size 2×2 or larger (that is, not completely diagonal) is defective. (A diagonal matrix is a special case of the Jordan normal form and is not defective.) For example, the n × n Jordan block,

has an eigenvalue, λ, with algebraic multiplicity n, but only one distinct eigenvector,

In fact, any defective matrix has a nontrivial Jordan normal form, which is as close as one can come to diagonalization of such a matrix.

## Example

A simple example of a defective matrix is:

which has a double eigenvalue of 3 but only one distinct eigenvector

(and constant multiples thereof).

## See also

## Notes

- Golub & Van Loan (1996, p. 316)

## References

- Golub, Gene H.; Van Loan, Charles F. (1996),
*Matrix Computations*(3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 - Strang, Gilbert (1988).
*Linear Algebra and Its Applications*(3rd ed.). San Diego: Harcourt. ISBN 978-970-686-609-7.