In Section 6.1, we will be concerned with the equation $A\mathbf{x}=\lambda \mathbf{x}$. This equation occurs in many applications of linear algebra. If the equation has a nonzero solution x, then $\lambda$ is said to be an eigenvalue of A and x is said to be an eigenvector belonging to $\lambda$.

Eigenvalues are a common part of our life whether we realize it or not. Wherever there are vibrations, there are eigenvalues, the natural frequencies of the vibrations. If you have ever tuned a guitar, you have solved an eigenvalue problem. When engineers design structures, they are concerned with the frequencies of vibration of the structure. This concern is particularly important in earthquake-prone regions such as California. The eigenvalues of a boundary value problem can be used to determine the energy states of an atom or critical loads that cause buckling in a beam. This latter application is presented in Section 6.1.

In Section 6.2, we will learn more about how to use eigenvalues and eigenvectors to solve systems of linear differential equations. We will consider a number of applications, including mixture problems, the harmonic motion of a system of springs, and the vibrations of a building. The motion of a building can be modeled by a second-order system of differential equations of the form

where $\mathbf{Y}\left(t\right)$ is a vector whose entries are all functions of t and ${\mathbf{Y}}^{″}\left(t\right)$ is the vector of functions formed by taking the second derivatives of each of the entries of Y(t). The solution of the equation is determined by the eigenvalues and eigenvectors of the matrix $A=M^{-1}K$.

In general, we can view eigenvalues as natural frequencies associated with linear operators. If A is an $n\times n$ matrix, we can think of A as representing a linear operator on ${ℝ}^n$. Eigenvalues and eigenvectors provide the key to understanding how the operator works. For example, if $\lambda$ > 0, the effect of the operator on any eigenvector belonging to $\lambda$ is simply a stretching or shrinking by a constant factor. Indeed, the effect of the operator is easily determined on any linear combination of eigenvectors. In particular, if it is possible to find a basis of eigenvectors for ${ℝ}^n$, the operator can be represented by a diagonal matrix D with respect to that basis and the matrix A can be factored into a product $XD{X}^{-1}$. In Section 6.3, we see how this is done and look at a number of applications.

In Section 6.4, we consider matrices with complex entries. In this setting, we will be concerned with matrices whose eigenvectors can be used to form an orthonormal basis for ${ℂ}^n$ (the vector space of all n-tuples of complex numbers). In Section 6.5, we introduce the singular value decomposition of a matrix and show four applications. Another important application of this factorization will be presented in Chapter 7.

Section 6.6 deals with the application of eigenvalues to quadratic equations in several variables and also with applications involving maxima and minima of functions of several variables. In Section 6.7, we consider symmetric positive definite matrices. The eigenvalues of such matrices are real and positive. These matrices occur in a wide variety of applications. Finally, in Section 6.8, we study matrices with nonnegative entries and some applications to economics.