Defective matrix

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

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

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

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

A simple example of a defective matrix is

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

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

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

(and constant multiples thereof).

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

• 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.