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 ${\displaystyle n\times n}${\displaystyle n\times n} matrix is defective if and only if it does not have ${\displaystyle n}${\displaystyle 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.

An ${\displaystyle n\times n}${\displaystyle n\times n} defective matrix always has fewer than ${\displaystyle n}${\displaystyle n} distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues ${\displaystyle \lambda }${\displaystyle \lambda } with algebraic multiplicity ${\displaystyle m>1}${\displaystyle m>1} (that is, they are multiple roots of the characteristic polynomial), but fewer than ${\displaystyle m}${\displaystyle m} linearly independent eigenvectors associated with ${\displaystyle \lambda }${\displaystyle \lambda }. If the algebraic multiplicity of ${\displaystyle \lambda }${\displaystyle \lambda } exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with ${\displaystyle \lambda }${\displaystyle \lambda }), then ${\displaystyle \lambda }${\displaystyle \lambda } is said to be a defective eigenvalue.^[1] However, every eigenvalue with algebraic multiplicity ${\displaystyle m}${\displaystyle m} always has ${\displaystyle m}${\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.

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

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

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

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

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

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

• Linear algebra
• Matrices (mathematics)

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