Chapter 3
Eigenvalue Problem
Solutions of systems of linear algebraic equations are briefly reviewed and used to
introduce the eigenvalue problem for square matrices. Properties of the eigenvalues
and eigenvectors for symmetric real-valued matrices are first considered. These
properties are then extended to real-valued nonsymmetric matrices.
The properties of the eigenvalues and eigenvectors of symmetric and nonsymmetric matrices in this chapter and the methods for analyzing SDOF systems in
Chap. 2 are used to calculate the dynamical response of MDOF and continuous
systems in Chaps. 4 and 5.
3.1 Systems of Linear Algebraic Equations
Consider the linear system of algebraic equations a x = b, where a is an (n, n)matrix and b is an (n, 1)-matrix, i.e., an n-dimensional column vector. It is assumed
that the entries of the matrix a and the vector b are real numbers.
The system of equations is said to be inhomogeneous if the right side is not
zero, i.e., b = 0, and homogeneous otherwise, i.e., b = 0. The solutions of these
two types of systems differ significantly.
– Inhomogeneous systems: If the inverse a −1 of a exists, the system of equations
has a unique solution given by x = a −1 b.
– Homogeneous systems: The trivial solution, i.e., the null vector x = 0, is always
a solution. However, if det(a) = 0, the trivial solution is not the only solution.
Under this condition, the homogeneous system of equations a x = 0 admits nontrivial solutions x = 0 since its equations are not linearly independent, i.e., one
or more equations are linear forms of the other equations.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_3
55
Eigenvalue Problem
Solutions of systems of linear algebraic equations are briefly reviewed and used to
introduce the eigenvalue problem for square matrices. Properties of the eigenvalues
and eigenvectors for symmetric real-valued matrices are first considered. These
properties are then extended to real-valued nonsymmetric matrices.
The properties of the eigenvalues and eigenvectors of symmetric and nonsymmetric matrices in this chapter and the methods for analyzing SDOF systems in
Chap. 2 are used to calculate the dynamical response of MDOF and continuous
systems in Chaps. 4 and 5.
3.1 Systems of Linear Algebraic Equations
Consider the linear system of algebraic equations a x = b, where a is an (n, n)matrix and b is an (n, 1)-matrix, i.e., an n-dimensional column vector. It is assumed
that the entries of the matrix a and the vector b are real numbers.
The system of equations is said to be inhomogeneous if the right side is not
zero, i.e., b = 0, and homogeneous otherwise, i.e., b = 0. The solutions of these
two types of systems differ significantly.
– Inhomogeneous systems: If the inverse a −1 of a exists, the system of equations
has a unique solution given by x = a −1 b.
– Homogeneous systems: The trivial solution, i.e., the null vector x = 0, is always
a solution. However, if det(a) = 0, the trivial solution is not the only solution.
Under this condition, the homogeneous system of equations a x = 0 admits nontrivial solutions x = 0 since its equations are not linearly independent, i.e., one
or more equations are linear forms of the other equations.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_3
55