Importance of eigenvalues

One practical application that comes to my mind is principal component analysis (PCA). It is extensively used in pattern recognition, for eg, face recognition. The covariance matrix of the observed data is used for this purpose. Let $C$ be our $N \times N$ covariance matrix, then \begin{align} C=\sum_{i}^{N}\lambda_iv_iv_i^H \end{align} is the eigen decomposition. Here $\lambda_i$ are the eigenvalues and $v_i$ are the eigenvectors. For a covariances matrix, $\lambda_i$ are always non-negative. Let us say $\lambda_1,\lambda_2,...\lambda_N$ are in the decreasing order. Thus, in that sense, $\lambda_1v_1v_1^H$ is the principal component in the above summation ahd hence plays a important role in determining the observed data. This interpretation will be more clear, once you delve into the application. My area is signal processing, and I see them Left, Right and Center throughout on my way. Happy Learning!!

The reason why eigenvalues are so important in mathematics are too many. Here is a very short and extremely incomplete list of the main applications I encountered in my path and that are coming now in mind to me:

Theoretical applications:

1. The eigenvalues of the Jacobian of a vector field at a given point determines the local geometry of the flow and the stability of that point;
2. An iterative method $\mathbf{y}_{k+1} = \mathbf{A} \mathbf{y}_k$ is convergent if the spectral radius $\rho(\mathbf{A})$ (the maximum absolute value of the eigenvalues of $\mathbf{A}$) is < 1.

Practical applications:

1. The order in which your search results appear in Google is determined by computing an eigenvector (See PageRank).
2. You can automatically recognize faces by computing eigenvectors of images (See Eigenfaces).

And there are also other type of eigenvalue problems, more difficult to solve, e.g. Generalized and Nonlinear Eigenvalue Problems, with even more interesting applications.

Hope I gave you some interesting inputs :)

Construction of suspension bridges... Each such bridge has a "natural frequency" and when the physical system that models the bridge is linearized, this frequency corresponds to the eigenvalue of smallest magnitude. Engineers want that this frequency be reasonably far away from any frequencies that occur naturally, such as local wind conditions. When this fails to be the case, i.e. when the natural frequency of the bridge corresponds to wave frequencies induced by the environment, the frequencies become (nearly) additive, as opposed to the desired state of the frequences (nearly) canceling out. The result is, for example, what occurred in the 1940 collapse of the Tacoma Narrows Bridge.