3.2 Eigenvalue Problem
57
1. The null vector x = 0 is not a right/left eigenvector or eigenvector although it
satisfies the eigenvalue equation a x = λ x or a T x = λ x. However, it violates
the first requirement of the above definitions.
2. The right/left eigenvectors of nonsymmetric matrices and the eigenvectors of
symmetric matrices can be calculated up to a multiplicative constant. For
example, let x k be a right eigenvector corresponding to the eigenvalue λ k , so that
x k = 0 and a x k = λ k x k . This system of equations multiplied by an arbitrary
scalar β becomes a (β x k ) = λ k (β x k ) so that β x k is also a right eigenvector of
a corresponding to the same eigenvalue λ k .
3. We refer to x k in the above definition as a right/left eigenvector or eigenvector of
a since eigenvalues can have multiple eigenvectors. For example, the (2,2)-matrix
a =
2 0
3 2
⇒ det
2 − λ 0
3
2− λ
= (λ − 2)
2
= 0
has a single eigenvalue (λ 1 = λ 2 = 2) but two right/left eigenvectors (see the
subsequent subsection and Appendix D).
4. The eigenvalues of nonsymmetric matrices can be real- and/or complex-valued
and so are the right/left eigenvectors.
Calculation of Eigenvalues/Eigenvectors The hand calculation of eigenvalues
and eigenvectors is feasible for small matrices. The following two-step approach
can be used.
– Step 1. Eigenvalues: Find the roots {λ k }, k = 1, . . . , n, of the polynomial det
a−
λ I
, i.e., the solutions of
det
a − λ I
= det
a
T
− λ I
= 0.
(3.1)
The roots are the eigenvalues of matrices a and a T . They may or may not be
distinct. They are real- and/or complex-valued depending on a.
– Step 2. Eigenvectors: The non-trivial solutions {x k }, k = 1, . . . , n, of the
homogeneous systems of equations
a − λ k I
x k = 0 ⇐⇒ a x k = λ k x k
(Right eigenvectors)
a
T
− λ k I
x k = 0 ⇐⇒ a
T x k = λ k x k (Left eigenvectors)
(3.2)
are the right and left eigenvectors of a. As previously stated, these systems of
equations admit non-trivial solutions since the determinants of
a−λ I
and
a T −
λ I
are zero for λ = λ k , k = 1, . . . , n. To avoid confusion, we will denote the
right and left eigenvectors corresponding to an eigenvalue λ k by u k and v k . If a is
symmetric, i.e., a = a T , the above equations coincide and so do their solutions,
i.e., the right and left eigenvectors. The non-trivial solutions of either equation in
Eq. 3.2 give the eigenvectors of a.
57
1. The null vector x = 0 is not a right/left eigenvector or eigenvector although it
satisfies the eigenvalue equation a x = λ x or a T x = λ x. However, it violates
the first requirement of the above definitions.
2. The right/left eigenvectors of nonsymmetric matrices and the eigenvectors of
symmetric matrices can be calculated up to a multiplicative constant. For
example, let x k be a right eigenvector corresponding to the eigenvalue λ k , so that
x k = 0 and a x k = λ k x k . This system of equations multiplied by an arbitrary
scalar β becomes a (β x k ) = λ k (β x k ) so that β x k is also a right eigenvector of
a corresponding to the same eigenvalue λ k .
3. We refer to x k in the above definition as a right/left eigenvector or eigenvector of
a since eigenvalues can have multiple eigenvectors. For example, the (2,2)-matrix
a =
2 0
3 2
⇒ det
2 − λ 0
3
2− λ
= (λ − 2)
2
= 0
has a single eigenvalue (λ 1 = λ 2 = 2) but two right/left eigenvectors (see the
subsequent subsection and Appendix D).
4. The eigenvalues of nonsymmetric matrices can be real- and/or complex-valued
and so are the right/left eigenvectors.
Calculation of Eigenvalues/Eigenvectors The hand calculation of eigenvalues
and eigenvectors is feasible for small matrices. The following two-step approach
can be used.
– Step 1. Eigenvalues: Find the roots {λ k }, k = 1, . . . , n, of the polynomial det
a−
λ I
, i.e., the solutions of
det
a − λ I
= det
a
T
− λ I
= 0.
(3.1)
The roots are the eigenvalues of matrices a and a T . They may or may not be
distinct. They are real- and/or complex-valued depending on a.
– Step 2. Eigenvectors: The non-trivial solutions {x k }, k = 1, . . . , n, of the
homogeneous systems of equations
a − λ k I
x k = 0 ⇐⇒ a x k = λ k x k
(Right eigenvectors)
a
T
− λ k I
x k = 0 ⇐⇒ a
T x k = λ k x k (Left eigenvectors)
(3.2)
are the right and left eigenvectors of a. As previously stated, these systems of
equations admit non-trivial solutions since the determinants of
a−λ I
and
a T −
λ I
are zero for λ = λ k , k = 1, . . . , n. To avoid confusion, we will denote the
right and left eigenvectors corresponding to an eigenvalue λ k by u k and v k . If a is
symmetric, i.e., a = a T , the above equations coincide and so do their solutions,
i.e., the right and left eigenvectors. The non-trivial solutions of either equation in
Eq. 3.2 give the eigenvectors of a.