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Eigenvectors and eigenvalues are used widely in science and engineering.

eigenvalue problem is a problem of considerable theoretical interest and wide-ranging
application. For example, this problem is crucial in solving systems of differential
equations, analyzing population growth models, and calculating powers of matrices (in
order to define the exponential matrix). Other areas such as physics, sociology, biology,
economics and statistics have focused considerable attention on "eigenvalues" and
"eigenvectors"-their applications and their computations
Eigenvectors and eigenvalues have many applications, particularly in physics.
Consider rigid physical bodies. Rigid physical bodies have a preferred direction of
rotation, about which they can rotate freely. For example, if someone were to throw a
football it would rotate around its axis while flying prettily through the air. If someone
were to hit the ball in the air, the ball would be likely to flop in a less simple way.
Although this may seem like common sense, even rigid bodies with a more complicated
shape will have preferred directions of rotation. These are called axes of inertia, and they
are calculated by finding the eigenvectors of a matrix called the inertia tensor. The
eigenvalues, also important, are called moments of inertia.
The major reason for studying eigenvalues and eigenvectors is that they are used in
many important mathematical results. In structural engineering, Eigenvalues and
Eigenvectors are used in dynamic analysis of structures, First we calculate dynamics
equation of the structure, during solving these equations we get eigenvalues as natural
frequency of the structure, eigenvectors gives the mode shapes of the structure. For
example least value of the eigenvalues gives the fundamental frequency of the structure,
corresponding first mode shape of the structure. Mode shape gives relative displacement
of the structure.Perhaps,by showing one of the example of a real world application may
make us more understand the importance of learning eigenvalues and eigenvectors.
On July 1, 1940, the Tacoma Narrows Bridge opened in Washington state. It
connected the city of Tacoma with the Kitsap Penninsula and ran over the Tacoma
Narrows which is a strait across the Puget Sound. Four months after it was built, it
collapsed. This collapse can be explained in engineering terms using the idea
One explanation centers around natural frequencies. The natural frequency is "the
frequency at which a system naturally vibrates once it has been set into motion". In other
words, the natural frequency is the characteristic motion of structure. It's the motion that a
structure takes on in response to wind or being walked on. It is especially important in the
design of musical instruments and in the tuning of radios.
Mathematically, the natural frequency can be characterized by the eigenvalue of the
smallest magnitude.
The model suggests that the "oscillations of the bridge were caused by the frequency
of the wind being too close to the natural frequency of the bridge." When frequencies
match, they compound which proved too strong a force for the bridge.

Many applications of matrices in both engineering and science utilize eigenvalues

and, some-times, eigenvectors. Control theory, vibration analysis, electric circuits,
advanced dynamics and quantum mechanics are just a few of the application areas.
One of the application that going to discussed here is the vibration analysis. The
eigenvalues are used to determine the natural frequencies (or eigen-frequencies) of
vibration, and the eigenvectors determine the shapes of these vibrational modes. This is a
practical applications in structural engineering. The eigenvalues of the system represent
the square of the natural frequencies with which the system will vibrate and the
eigenvectors represent the mode shapes. In other word, the eigenvectors denote how the
masses vibrate with respect to each other.
Most structures from buildings to bridges have a natural frequency of vibration. It
means all these structures have their own system of eigen-vibrations and eigenfrequencies. Now external forces like wind and earthquake may cause these structures to
undergo vibrations. In case the frequency of these vibrations becomes equal to the natural
frequencies of these structures, vibrations with large amplitudes are set up. It is a
phenomena called resonance. This can lead to the collapse of the structure by a process
called aeroelastic flutter. One very famous example of the collapse has been discussed
Eigenvalues can also be used to test for cracks or deformities in structural
components used for construction. When a beam is struck, its natural frequencies
(eigenvalues) can be heard or measured. If the beam "rings," then it is not flawed. A dull
sound will result from a flawed beam because the flaw causes the eigenvalues to change.
Sensitive machines can be used to "see" and "hear" eigenvalues more precisely.
The eigenvalues can also be used to determine if a structure has deformed under the
application of a particular force. Eigenvalues for the structure are measured before and
after the application of force. If a change in the eigenvalues is observed, it means the
structure has undergone deformation.
Eigenvalues and Eigenvectors could also be used for modal analysis. In this case a
big model would be converted to a number of single degree freedom springs. Then
loading is also passed through a similar transformation. Time history could then be
applied to a series of single degree freedom springs and the result are added up again
using reverse of the transformation used before. This is much faster than full model time
history analysis. However, with the speed of computers these days, this method is not
used that often.
Eigenvalue and Eigenvectors are also used for buckling calculations. For such type
of analyses you also need to use different member stiffness matrices to incorporate the
axial loads. In buckling analysis, an initial load is applied to the structural model, then the
buckling Eigenvalue indicates the number of times this load has to be increased for a
buckling to take place. The eigenvector shows the resulting shape of the buckling mode.