- Source: 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
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
References
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.