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 {\displaystyle n\times n} matrix is defective if and only if it does not have n {\displaystyle n} 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 n × n {\displaystyle n\times n} defective matrix always has fewer than n {\displaystyle n} distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ {\displaystyle \lambda } with algebraic multiplicity m > 1 {\displaystyle m>1} (that is, they are multiple roots of the characteristic polynomial), but fewer than m {\displaystyle m} linearly independent eigenvectors associated with λ {\displaystyle \lambda } . If the algebraic multiplicity of λ {\displaystyle \lambda } exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with λ {\displaystyle \lambda } ), then λ {\displaystyle \lambda } is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity m {\displaystyle m} always has 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.
Jordan block
Any nontrivial Jordan block of size 2 × 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 1 × 1 {\displaystyle 1\times 1} and is not defective.) For example, the n × n {\displaystyle n\times n} Jordan block
J = [ λ 1 λ ⋱ ⋱ 1 λ ] , {\displaystyle J={\begin{bmatrix}\lambda &1&\;&\;\\\;&\lambda &\ddots &\;\\\;&\;&\ddots &1\\\;&\;&\;&\lambda \end{bmatrix}},}has an eigenvalue, λ {\displaystyle \lambda } with algebraic multiplicity n {\displaystyle n} (or greater if there are other Jordan blocks with the same eigenvalue), but only one distinct eigenvector J v 1 = λ v 1 {\displaystyle Jv_{1}=\lambda v_{1}} , where v 1 = [ 1 0 ⋮ 0 ] . {\displaystyle v_{1}={\begin{bmatrix}1\\0\\\vdots \\0\end{bmatrix}}.} The other canonical basis vectors v 2 = [ 0 1 ⋮ 0 ] , … , v n = [ 0 0 ⋮ 1 ] {\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 J v k = λ v k + v k − 1 {\displaystyle Jv_{k}=\lambda v_{k}+v_{k-1}} for k = 2 , … , 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.
Example
A simple example of a defective matrix is
[ 3 1 0 3 ] , {\displaystyle {\begin{bmatrix}3&1\\0&3\end{bmatrix}},}which has a double eigenvalue of 3 but only one distinct eigenvector
[ 1 0 ] {\displaystyle {\begin{bmatrix}1\\0\end{bmatrix}}}(and constant multiples thereof).
See also
- Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities
Notes
- 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.
References
Golub & Van Loan (1996, p. 316) - Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 ↩
Golub & Van Loan (1996, p. 316) - Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9 ↩