Eigenvalues and Eigenvectors
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).
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 ...