Appendix D
Generalized Eigenvectors
The representation of the solution of MDOF systems as elements of the linear
spaces spanned by modal shapes (proportional damping) or right/left eigenvectors
(non-proportional damping) is possible if the eigenvalues are distinct so that the
corresponding eigenvectors span the solution spaces. If one or more eigenvalues are
multiple, the corresponding set of eigenvectors is insufficient to span the solution
space. It has to be augmented by additional vectors, referred to as generalized
eigenvectors.
Definition 4 Consider an (n, n)-matrix a with an eigenvalue λ 1 of multiplicity k.
The n-dimensional vector x k defined by
a − λ 1 I
(k−1) x j = 0 and
a − λ 1 I
(k) x k = 0
(D.1)
is called generalized eigenvector or eigenvector of rank k of the (n, n)-matrix a
associated with the eigenvalue λ 1 .
Definition 5 Consider, as above, an (n, n)-matrix a with an eigenvalue λ 1 of
multiplicity k. Define the vectors x j , j = 1, . . . , k − 1, by the following conditions:
a − λ 1 I
x k = x k−1 ,
a − λ 1 I
(2) x k = x k−2 , . . . ,
a − λ 1 I
(k−1) x k = x 1 .
(D.2)
These vectors have two notable properties.
Property D.1 The vectors x j , j = 1, . . . , k − 1, are generalized eigenvectors of
rank j , which means that
a − λ 1 I
(j −1) x j = 0 and
a − λ 1 i
(j ) x j = 0.
Proof We have x j =
a − λ 1 i
(k−j) x k from Eq. D.2 so that
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
147
Generalized Eigenvectors
The representation of the solution of MDOF systems as elements of the linear
spaces spanned by modal shapes (proportional damping) or right/left eigenvectors
(non-proportional damping) is possible if the eigenvalues are distinct so that the
corresponding eigenvectors span the solution spaces. If one or more eigenvalues are
multiple, the corresponding set of eigenvectors is insufficient to span the solution
space. It has to be augmented by additional vectors, referred to as generalized
eigenvectors.
Definition 4 Consider an (n, n)-matrix a with an eigenvalue λ 1 of multiplicity k.
The n-dimensional vector x k defined by
a − λ 1 I
(k−1) x j = 0 and
a − λ 1 I
(k) x k = 0
(D.1)
is called generalized eigenvector or eigenvector of rank k of the (n, n)-matrix a
associated with the eigenvalue λ 1 .
Definition 5 Consider, as above, an (n, n)-matrix a with an eigenvalue λ 1 of
multiplicity k. Define the vectors x j , j = 1, . . . , k − 1, by the following conditions:
a − λ 1 I
x k = x k−1 ,
a − λ 1 I
(2) x k = x k−2 , . . . ,
a − λ 1 I
(k−1) x k = x 1 .
(D.2)
These vectors have two notable properties.
Property D.1 The vectors x j , j = 1, . . . , k − 1, are generalized eigenvectors of
rank j , which means that
a − λ 1 I
(j −1) x j = 0 and
a − λ 1 i
(j ) x j = 0.
Proof We have x j =
a − λ 1 i
(k−j) x k from Eq. D.2 so that
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
147