Numerical Methods for Engineers and Scientists 3rd Edition

Chapter 5

Eigenvalues and Eigenvectors

Core Topics

The characteristic Equation (5.2)

Basic power method (5.3).

Inverse power method (5.4).

Shifted power method (5.5).

QR factorization and iteration method (5.6).

Use of MATLAB's built-in functions for determining eigenvalues and eigenvectors (5.7).

5.1 BACKGROUND

For a given (n × n) matrix [a], the number λ is an eigenvalue¹ of the matrix if:

The vector [u] is a column vector with n elements called the eigenvector, associated with the eigenvalue λ.

Equation (5.1) can be viewed in a more general way. The multiplication [a][u] is a mathematical operation and can be thought of as the matrix [a] operating on the operand [u]. With this terminology, Eq. (5.1) can be read as “[a] operates on [u] to yield λ times [u],” and Eq. (5.1) can be generalized to any mathematical operation as:

where L is an operator that can represent multiplication by a matrix, differentiation, integration, and so on, u is a vector or function, and λ is a scalar constant. For example, if L represents second differentiation with respect to x, y is a function of x, and k is a constant, then Eq. (5.2) can have the form:

Equation (5.2) is a general statement of an ...