64
3 Eigenvalue Problem
are the right and left eigenvectors of a which can be obtained by calling the
MATLAB function in Eq. 3.3.
Proof The conditions of Eq. 3.2 with the notations u and v for the right and left
eigenvectors become
a u k = λ k u k and a
T v l = λ l v l .
(3.9)
The first equation gives v T
l a u k = λ k v T
l u k by left multiplication with v T
l .
The transposed of the second equation becomes v T
l a u k = λ l v T
l u k by right
multiplication with u k . The left sides of the resulting two equations coincide so
that their right sides must coincide, i.e., λ k v T
l u k = λ l v T
l u k or
λ k −λ l
v T
l u k =
0. Since λ k = λ l by assumption, we have v T
l u k = 0. The above equalities also
give v T
l a u k = 0, k = l. The matrix form of the orthogonality condition in
Eq. 3.8 results by direct calculations. The reader is encouraged to perform these
calculations since orthogonality is used extensively in the latter part of this book.
Note also that the equality λ k v T
l u k = λ l v T
l u k above holds for any k and l.
For k = l, we have
λ k =
v T
k a u k
v T
k u k
, i = 1, . . . , n,
(3.10)
which constitutes a useful relationship for the dynamic analysis of MDOF
systems.
3. If the eigenvalues are distinct, the right eigenvectors and the left eigenvectors
define bases of R n .
Proof It is shown in Appendix C that the sets of right and left eigenvectors
are linearly independent. The orthogonality property of Eq. 3.8 can be used to
calculate the components of arbitrary vectors of R n . An arbitrary element x
of R n admits the representation x =
n
i=1 x, u i u i , where x, u i denoted
the projection of x on u i (see Appendix C). The vector x admits a similar
representation in the basis defined by the left eigenvectors of a.
4. Nonsymmetric matrices can be positive definite.
Proof For example, the (2, 2)-nonsymmetric matrix
a =
2 0
2 2
⇒ x
T a x = (x 1 + x 2 )
2
+ x
2
1 + x
2
2 > 0
is positive definite. We do not explore further properties of these types of matrices
since they are not used in our discussion. The interest reader can consult [1]
(Sect. 5.3).
3 Eigenvalue Problem
are the right and left eigenvectors of a which can be obtained by calling the
MATLAB function in Eq. 3.3.
Proof The conditions of Eq. 3.2 with the notations u and v for the right and left
eigenvectors become
a u k = λ k u k and a
T v l = λ l v l .
(3.9)
The first equation gives v T
l a u k = λ k v T
l u k by left multiplication with v T
l .
The transposed of the second equation becomes v T
l a u k = λ l v T
l u k by right
multiplication with u k . The left sides of the resulting two equations coincide so
that their right sides must coincide, i.e., λ k v T
l u k = λ l v T
l u k or
λ k −λ l
v T
l u k =
0. Since λ k = λ l by assumption, we have v T
l u k = 0. The above equalities also
give v T
l a u k = 0, k = l. The matrix form of the orthogonality condition in
Eq. 3.8 results by direct calculations. The reader is encouraged to perform these
calculations since orthogonality is used extensively in the latter part of this book.
Note also that the equality λ k v T
l u k = λ l v T
l u k above holds for any k and l.
For k = l, we have
λ k =
v T
k a u k
v T
k u k
, i = 1, . . . , n,
(3.10)
which constitutes a useful relationship for the dynamic analysis of MDOF
systems.
3. If the eigenvalues are distinct, the right eigenvectors and the left eigenvectors
define bases of R n .
Proof It is shown in Appendix C that the sets of right and left eigenvectors
are linearly independent. The orthogonality property of Eq. 3.8 can be used to
calculate the components of arbitrary vectors of R n . An arbitrary element x
of R n admits the representation x =
n
i=1 x, u i u i , where x, u i denoted
the projection of x on u i (see Appendix C). The vector x admits a similar
representation in the basis defined by the left eigenvectors of a.
4. Nonsymmetric matrices can be positive definite.
Proof For example, the (2, 2)-nonsymmetric matrix
a =
2 0
2 2
⇒ x
T a x = (x 1 + x 2 )
2
+ x
2
1 + x
2
2 > 0
is positive definite. We do not explore further properties of these types of matrices
since they are not used in our discussion. The interest reader can consult [1]
(Sect. 5.3).