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 matrix is defective if and only if it does not have 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 defective matrix always has fewer than distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues with algebraic multiplicity , but fewer than linearly independent eigenvectors associated with. If the algebraic multiplicity of exceeds its geometric multiplicity, then is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity always has linearly independent generalized eigenvectors.
A real symmetric matrix and more generally a Hermitian matrix, and a unitary matrix, is never defective; more generally, a normal matrix is never defective.
Jordan block
Any nontrivial Jordan block of size or larger is defective. For example, the Jordan blockhas an eigenvalue, with algebraic multiplicity , but only one distinct eigenvector, where The other canonical basis vectors form a chain of generalized eigenvectors such that for.
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 iswhich has a double eigenvalue of 3 but only one distinct eigenvector
.