56
3 Eigenvalue Problem
Example 3.1 The determinant of the (2, 2)-matrix
a =
2 3
4 6
is det(a) = (2)(6) − (3)(4) = 0 so that we expect that the homogeneous system
a x = 0 admits non-trivial solutions. To see this, set x 1 = c = 0, where c is an
arbitrary non-zero constant. From the first equation, we have x 2 = −2 c/3. Since
x 1 = c and x 2 = −2 c/3 also satisfies the second equation, the system has the
infinite set of non-trivial solutions x T = [c − 2 c/3] indexed by c, which can be
any non-zero real number. Note that the second equation 4 x 1 + 6 x 2 = 0 of the
homogeneous system a x = 0 is the first equation, 2 x 1 + 3 x 2 = 0, multiplied by
2. The two equations of the system are linearly dependent (see Appendix C).
3.2 Eigenvalue Problem
Consider an (n, n)-matrix a and the system of linear algebraic equations a x = λ x,
where λ is a scalar. An alternative form of this system is
a − λ I
x = 0, where
I denotes the (n, n)-identity matrix, i.e., an (n, n)-diagonal matrix with unit nonzero entries. We have seen that this system of equations admits non-trivial solutions
if the determinant of the matrix
a − λ I
is zero. This condition is satisfied by the
roots of the n-degree polynomial det
a−λ I
. We limit the discussion to real-valued
matrices a, i.e., matrices a whose entries are reals.
Definition 3.1 The roots λ 1 , . . . , λ n of the n-degree polynomial det
a − λ I
, i.e.,
the solutions of det
a − λ I
= 0, are called the eigenvalues of a. Since the
determinant of a matrix and its transpose coincide, e.g., the determinants of matrices
a − λ I
and
a − λ I
T =
a T − λ I
, the matrix a and its transpose a T have the
same eigenvalues. The eigenvalues λ 1 , . . . , λ n may or may not be distinct and can
be real or complex.
Definition 3.2 Let λ k be an eigenvalue of an (n, n) real-valued matrix a or,
equivalently, of its transpose a T . An n-dimensional vector x k is a right eigenvector
of this matrix corresponding to λ k if (1) it is not the null vector, i.e., x k = 0, and
(2) it satisfies the linear system of equations a x k = λ k x k . An n-dimensional vector
x k is a left eigenvector of matrix a corresponding to λ k if (1) it is not the null
vector, i.e., x k = 0, and (2) it satisfies the linear system of equations a T x k = λ k x k .
If a is symmetric, the right and left eigenvectors coincide, and are referred to as
eigenvectors of a.
We list properties of the eigenvalues and eigenvectors of symmetric and nonsymmetric matrices which result directly from the above definitions. They are useful for
numerical applications and analytical derivations.
3 Eigenvalue Problem
Example 3.1 The determinant of the (2, 2)-matrix
a =
2 3
4 6
is det(a) = (2)(6) − (3)(4) = 0 so that we expect that the homogeneous system
a x = 0 admits non-trivial solutions. To see this, set x 1 = c = 0, where c is an
arbitrary non-zero constant. From the first equation, we have x 2 = −2 c/3. Since
x 1 = c and x 2 = −2 c/3 also satisfies the second equation, the system has the
infinite set of non-trivial solutions x T = [c − 2 c/3] indexed by c, which can be
any non-zero real number. Note that the second equation 4 x 1 + 6 x 2 = 0 of the
homogeneous system a x = 0 is the first equation, 2 x 1 + 3 x 2 = 0, multiplied by
2. The two equations of the system are linearly dependent (see Appendix C).
3.2 Eigenvalue Problem
Consider an (n, n)-matrix a and the system of linear algebraic equations a x = λ x,
where λ is a scalar. An alternative form of this system is
a − λ I
x = 0, where
I denotes the (n, n)-identity matrix, i.e., an (n, n)-diagonal matrix with unit nonzero entries. We have seen that this system of equations admits non-trivial solutions
if the determinant of the matrix
a − λ I
is zero. This condition is satisfied by the
roots of the n-degree polynomial det
a−λ I
. We limit the discussion to real-valued
matrices a, i.e., matrices a whose entries are reals.
Definition 3.1 The roots λ 1 , . . . , λ n of the n-degree polynomial det
a − λ I
, i.e.,
the solutions of det
a − λ I
= 0, are called the eigenvalues of a. Since the
determinant of a matrix and its transpose coincide, e.g., the determinants of matrices
a − λ I
and
a − λ I
T =
a T − λ I
, the matrix a and its transpose a T have the
same eigenvalues. The eigenvalues λ 1 , . . . , λ n may or may not be distinct and can
be real or complex.
Definition 3.2 Let λ k be an eigenvalue of an (n, n) real-valued matrix a or,
equivalently, of its transpose a T . An n-dimensional vector x k is a right eigenvector
of this matrix corresponding to λ k if (1) it is not the null vector, i.e., x k = 0, and
(2) it satisfies the linear system of equations a x k = λ k x k . An n-dimensional vector
x k is a left eigenvector of matrix a corresponding to λ k if (1) it is not the null
vector, i.e., x k = 0, and (2) it satisfies the linear system of equations a T x k = λ k x k .
If a is symmetric, the right and left eigenvectors coincide, and are referred to as
eigenvectors of a.
We list properties of the eigenvalues and eigenvectors of symmetric and nonsymmetric matrices which result directly from the above definitions. They are useful for
numerical applications and analytical derivations.