3.3 Problems
65
5. An eigenvalue λ 1 of multiplicity q ≥ 2 can be associated with q linearly
independent generalized right/left eigenvectors.
Proof See [1] (Sect. 6.3) for proof and technical considerations. The generalized
right eigenvectors can be obtained by the algorithm of Eq. 3.7. The generalized
left eigenvectors result from this equation with a T in place of a.
Example 3.6 The (2,2)-nonsymmetric matrix
a =
1 1
0 1
has the eigenvalues λ 1 = λ 2 = 1. The first right eigenvector x 1 given by MATLAB
has the components (1, 0). The second right eigenvector results from Eq. 3.7 with
q = 2, i.e., a u 2 = λ u 2 + u 1 which gives the equations u 2,1 + u 2,2 = u 2,1 + 1 and
u 2,2 = u 2,2 so that u 2,2 = 1 and u 2,1 is an arbitrary constant, where u k,r denotes
the rth component of u k .
3.3 Problems
Problem 3.1 Construct examples of linear homogeneous systems of algebraic
equations with n ≥ 3 which admit non-trivial solutions and find their solutions.
Problem 3.2 Consider an (n, n)-symmetric real-valued matrix a with eigenvectors
{x i } assumed to have unit length. Show that the (n, n)-matrix X T a X is diagonal
with non-zero entries {λ k }, where X = [x 1 x 2 . . . x n ].
Hint: Use the orthogonality conditions of Eq. 3.4.
Problem 3.3 Find the eigenvalues and the eigenvectors of the (2, 2)-matrix
a =
10 −3
−3 4
by hand calculations and by MATLAB. Compare results and plot the eigenvectors.
Problem 3.4 Find by hand calculations the eigenvalues and the right/left eigenvectors of the (2, 2)-matrix
a =
2 1
0 1
.
Show that they are orthogonal in the sense of Eq. 3.8.
65
5. An eigenvalue λ 1 of multiplicity q ≥ 2 can be associated with q linearly
independent generalized right/left eigenvectors.
Proof See [1] (Sect. 6.3) for proof and technical considerations. The generalized
right eigenvectors can be obtained by the algorithm of Eq. 3.7. The generalized
left eigenvectors result from this equation with a T in place of a.
Example 3.6 The (2,2)-nonsymmetric matrix
a =
1 1
0 1
has the eigenvalues λ 1 = λ 2 = 1. The first right eigenvector x 1 given by MATLAB
has the components (1, 0). The second right eigenvector results from Eq. 3.7 with
q = 2, i.e., a u 2 = λ u 2 + u 1 which gives the equations u 2,1 + u 2,2 = u 2,1 + 1 and
u 2,2 = u 2,2 so that u 2,2 = 1 and u 2,1 is an arbitrary constant, where u k,r denotes
the rth component of u k .
3.3 Problems
Problem 3.1 Construct examples of linear homogeneous systems of algebraic
equations with n ≥ 3 which admit non-trivial solutions and find their solutions.
Problem 3.2 Consider an (n, n)-symmetric real-valued matrix a with eigenvectors
{x i } assumed to have unit length. Show that the (n, n)-matrix X T a X is diagonal
with non-zero entries {λ k }, where X = [x 1 x 2 . . . x n ].
Hint: Use the orthogonality conditions of Eq. 3.4.
Problem 3.3 Find the eigenvalues and the eigenvectors of the (2, 2)-matrix
a =
10 −3
−3 4
by hand calculations and by MATLAB. Compare results and plot the eigenvectors.
Problem 3.4 Find by hand calculations the eigenvalues and the right/left eigenvectors of the (2, 2)-matrix
a =
2 1
0 1
.
Show that they are orthogonal in the sense of Eq. 3.8.