Defective matrix

An ${\displaystyle n\times n}$ defective matrix always has fewer than ${\displaystyle n}$ distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues ${\displaystyle \lambda }$ with algebraic multiplicity ${\displaystyle m>1}$ (that is, they are multiple roots of the characteristic polynomial), but fewer than ${\displaystyle m}$ linearly independent eigenvectors associated with ${\displaystyle \lambda }$. If the algebraic multiplicity of ${\displaystyle \lambda }$ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with ${\displaystyle \lambda }$), then ${\displaystyle \lambda }$ is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity ${\displaystyle m}$ always has ${\displaystyle m}$ 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 (which includes Hermitian and unitary matrices as special cases) is never defective.

Jordan block

Any nontrivial Jordan block of size ${\displaystyle 2\times 2}$ 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 ${\displaystyle 1\times 1}$ and is not defective.) For example, the ${\displaystyle n\times n}$ Jordan block

${\displaystyle J=\left[\begin{array}{cccc}\lambda & 1& & \\ & \lambda & \ddots & \\ & & \ddots & 1\\ & & & \lambda \end{array}\right],}$

has an eigenvalue, ${\displaystyle \lambda }$ with algebraic multiplicity ${\displaystyle n}$ (or greater if there are other Jordan blocks with the same eigenvalue), but only one distinct eigenvector ${\displaystyle J{v}_1=\lambda v_1}$, where ${\displaystyle v_1=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right].}$ The other canonical basis vectors ${\displaystyle v_2=\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right],\dots ,v_n=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right]}$ form a chain of generalized eigenvectors such that ${\displaystyle J{v}_k=\lambda v_k+v_{k-1}}$ for ${\displaystyle k=2,\dots ,n}$.

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