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,

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

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

$v = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}.$

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:

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

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

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

(and constant multiples thereof).

See also

• Jordan normal form

Notes

1. 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 0-8018-5414-8
• Strang, Gilbert (1988). Linear Algebra and Its Applications (3rd ed.). San Diego: Harcourt. ISBN 970-686-609-4.