Generalized eigenvector

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a generalized eigenvector of an 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> which satisfies certain criteria which are more relaxed than those for an (ordinary) <a href="/facts/Eigenvector/8TjEoT8u">eigenvector</a>.
Let 
 
 
 
 V
 
 
 {\displaystyle V}
 
 be an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> and let 
 
 
 
 A
 
 
 {\displaystyle A}
 
 be the <a href="/facts/Linear_map/Q1PBKNVV">matrix representation</a> of a linear map from 
 
 
 
 V
 
 
 {\displaystyle V}
 
 to 
 
 
 
 V
 
 
 {\displaystyle V}
 
 with respect to some ordered <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>.
There may not always exist a full set of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 <a href="/facts/Linear_independence/8JrHfIIa">linearly independent</a> eigenvectors of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 that form a complete basis for 
 
 
 
 V
 
 
 {\displaystyle V}
 
. That is, the matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 may not be <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalizable</a>. This happens when the <a href="/facts/Algebraic_multiplicity/8TjEoT8u">algebraic multiplicity</a> of at least one <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> 
 
 
 
 
 λ
 
 i
 
 
 
 
 {\displaystyle \lambda _{i}}
 
 is greater than its <a href="/facts/Geometric_multiplicity/8TjEoT8u">geometric multiplicity</a> (the <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">nullity</a> of the matrix 
 
 
 
 (
 A
 −
 
 λ
 
 i
 
 
 I
 )
 
 
 {\displaystyle (A-\lambda _{i}I)}
 
, or the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of its <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">nullspace</a>). In this case, 
 
 
 
 
 λ
 
 i
 
 
 
 
 {\displaystyle \lambda _{i}}
 
 is called a <a href="/facts/Defective_eigenvalue/I5uysqAs">defective eigenvalue</a> and 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is called a <a href="/facts/Defective_matrix/I5uysqAs">defective matrix</a>.
A generalized eigenvector 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 corresponding to 
 
 
 
 
 λ
 
 i
 
 
 
 
 {\displaystyle \lambda _{i}}
 
, together with the matrix 
 
 
 
 (
 A
 −
 
 λ
 
 i
 
 
 I
 )
 
 
 {\displaystyle (A-\lambda _{i}I)}
 
 generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an <a href="/facts/Invariant_subspace/51Vm1W2X">invariant subspace</a> of 
 
 
 
 V
 
 
 {\displaystyle V}
 
.
Using generalized eigenvectors, a set of linearly independent eigenvectors of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 can be extended, if necessary, to a complete basis for 
 
 
 
 V
 
 
 {\displaystyle V}
 
. This basis can be used to determine an "almost diagonal matrix" 
 
 
 
 J
 
 
 {\displaystyle J}
 
 in <a href="/facts/Jordan_normal_form/M7ezYbnl">Jordan normal form</a>, <a href="/facts/Matrix_similarity/ySevpav1">similar</a> to 
 
 
 
 A
 
 
 {\displaystyle A}
 
, which is useful in computing certain <a href="/facts/Matrix_function/T2e9sFm9">matrix functions</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
. The matrix 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is also useful in solving the <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">system of linear differential equations</a> 
 
 
 
 
 
 x
 
 ′
 
 =
 A
 
 x
 
 ,
 
 
 {\displaystyle \mathbf {x} '=A\mathbf {x} ,}
 
 where 
 
 
 
 A
 
 
 {\displaystyle A}
 
 need not be diagonalizable.
The dimension of the generalized eigenspace corresponding to a given eigenvalue 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is the algebraic multiplicity of 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
.

Generalized eigenvector open-in-new

Generalized eigenvector