# 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 eigenvalue^{1} 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 ...

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