2.2 The Algebraic EVP
Some familiarity with algebraic EVPs is required when dealing with differential
EVPs. For example, many methods for obtaining approximate solutions to differential EVPs require one to solve algebraic EVPs. The two types of EVPs share
many concepts, though the meaning of these may be different.
2.2.1 Mathematical Form
Algebraic EVPs take the form:
Ku ¼kLu;
ð2:1Þ
where u is an n-vector of state variables, K and L are n  n matrices with
components depending on the data of the system, and k is a scalar. All quantities
can be complex-valued.
Equation (2.1) is solved by u = 0, k 2 R. However, solving the algebraic EVP
means to find those values of k for which nontrivial solutions u 6 ¼ 0 exist. The
values of k allowing this are called eigenvalues, and the associated vectors u are
called eigen-vectors. In general there are n eigenvalues and eigenvectors of the
EVP. The pairs (k j , u j ), j = 1, n are called eigenpairs.
Example 2.1. In locating undamped natural frequencies and mode shapes for
multi-DOF systems, one is lead to solve the system of homogeneous equations
(K − x
2
M)u = 0, where K is the stiffness matrix, M the mass matrix, x
2 the
squared natural frequency, and u the mode shape (cf. (1.18)). This is an algebraic
EVP of the form (2.1), with x
2 being the eigenvalue.
2.2.2 Properties of Eigenvalues and Eigenvectors
For assessing questions of stability, it is often essential to know whether the
eigenvalues of a given EVP are real-valued. Here is a sufficient condition:
Theorem 2.1 : Real-valueness of eigenvalues. If the EVP (2.1) is
Hermetian, and if K is definite or K and L commute, then all eigenvalues k
are real.
Some definitions may be in order to appreciate this theorem: The EVP (2.1) is
Hermetian if K and L are both Hermetian matrices. A matrix K is Hermetian or selfadjoint if K ¼
K
T ; where the overbar denotes complex conjugation and K
T is the
50
2 Eigenvalue Problems of Vibrations and Stability
Some familiarity with algebraic EVPs is required when dealing with differential
EVPs. For example, many methods for obtaining approximate solutions to differential EVPs require one to solve algebraic EVPs. The two types of EVPs share
many concepts, though the meaning of these may be different.
2.2.1 Mathematical Form
Algebraic EVPs take the form:
Ku ¼kLu;
ð2:1Þ
where u is an n-vector of state variables, K and L are n  n matrices with
components depending on the data of the system, and k is a scalar. All quantities
can be complex-valued.
Equation (2.1) is solved by u = 0, k 2 R. However, solving the algebraic EVP
means to find those values of k for which nontrivial solutions u 6 ¼ 0 exist. The
values of k allowing this are called eigenvalues, and the associated vectors u are
called eigen-vectors. In general there are n eigenvalues and eigenvectors of the
EVP. The pairs (k j , u j ), j = 1, n are called eigenpairs.
Example 2.1. In locating undamped natural frequencies and mode shapes for
multi-DOF systems, one is lead to solve the system of homogeneous equations
(K − x
2
M)u = 0, where K is the stiffness matrix, M the mass matrix, x
2 the
squared natural frequency, and u the mode shape (cf. (1.18)). This is an algebraic
EVP of the form (2.1), with x
2 being the eigenvalue.
2.2.2 Properties of Eigenvalues and Eigenvectors
For assessing questions of stability, it is often essential to know whether the
eigenvalues of a given EVP are real-valued. Here is a sufficient condition:
Theorem 2.1 : Real-valueness of eigenvalues. If the EVP (2.1) is
Hermetian, and if K is definite or K and L commute, then all eigenvalues k
are real.
Some definitions may be in order to appreciate this theorem: The EVP (2.1) is
Hermetian if K and L are both Hermetian matrices. A matrix K is Hermetian or selfadjoint if K ¼
K
T ; where the overbar denotes complex conjugation and K
T is the
50
2 Eigenvalue Problems of Vibrations and Stability