Eigenvectors and eigenvalues come in pairs; that is why we use the term eigenpair. Some have asked me about practical applications of eigenpairs. So this post goes.

Did you know the connection between eigenvectors and Reggaeton Music (or music in general)? How about eigenvectors and bridges, car designers, speakers, architecture, or oil companies?

This tutorial from Rice University illustrates the importance of computing eigenpairs when it comes to determining optimum sound frequencies or solving engineering problems. Perhaps I should coin a new term: “Eiggaeton” (Eigenvectors + Reggaeton).

Let see some applications as mentioned in the Rice tutorial:

Eigenpairs and Bridges

“The interesting uses come from larger systems, so we will just discuss them rather than solve them. Have you ever seen the video of the collapse of the Tacoma Narrows Bridge? The Tacoma Bridge was built in 1940. From the beginning, the bridge would form small waves like the surface of a body of water. This accidental behavior of the bridge brought many people who wanted to drive over this moving bridge. Most people thought that the bridge was safe despite the movement. However, about four months later, the oscillations (waves) became bigger. At one point, one edge of the road was 28 feet higher than the other edge. Finally, this bridge crashed into the water below. One explanation for the crash is that the oscillations of the bridge were caused by the frequency of the wind being too close to the natural frequency of the bridge. The natural frequency of the bridge is the eigenvalue of smallest magnitude of a system that models the bridge. This is why eigenvalues are very important to engineers when they analyze structures.”

“Also, a bridge in Manchester, England collapsed in 1831 because of conflicts between frequencies. However, this time, the natural frequency of the bridge was matched by the frequency caused by soldiers marching in step. Large oscillations occurred and the bridge collapsed. This is why soldiers break cadence when crossing a bridge.”

“Frequencies are also used in electrical systems. When you tune your radio, you are changing the resonant frequency until it matches the frequency at which your station is broadcasting. Engineers used eigenvalues when they designed your radio.”

Eigenpairs and Music

“Frequencies are also vital in music performance. When instruments are tuned, their frequencies are matched. It is the frequency that determines what we hear as music. Although musicians do not study eigenvalues in order to play their instruments better, the study of eigenvalues can explain why certain sounds are pleasant to the ear while others sound “flat” or “sharp.” When two people sing in harmony, the frequency of one voice is a constant multiple of the other. That is what makes the sounds pleasant. Eigenvalues can be used to explain many aspects of music from the initial design of the instrument to tuning and harmony during a performance. Even the concert halls are analyzed so that every seat in the theater receives a high quality sound.”

Eigenpairs and Stereo Systems

“Car designers analyze eigenvalues in order to damp out the noise so that the occupants have a quiet ride. Eigenvalue analysis is also used in the design of car stereo systems so that the sounds are directed correctly for the listening pleasure of the passengers and driver. When you see a car that vibrates because of the loud booming music, think of eigenvalues. Eigenvalue analysis can indicate what needs to be changed to reduce the vibration of the car due to the music.”

Eigenpairs and Architecture

“If you were asked to build the strongest column that you could to support the weight of a roof using only a specified amount of material, what shape would that column take? Most of us would build a cylinder like most other columns that we have seen. However, Steve Cox of Rice University and Michael Overton of New York University proved, based on the work of J. Keller and I. Tadjbakhsh, that the column would be stronger if it was largest at the top, middle, and bottom. At the points  of the way from either end, the column could be smaller because the column would not naturally buckle there anyway.”

Eigenpairs and Fractures

“Eigenvalues can also be used to test for cracks or deformities in a solid. Can you imagine if every inch of every beam used in construction had to be tested? The problem is not as time consuming when eigenvalues are used. When a beam is struck, its natural frequencies (eigenvalues) can be heard. 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.”

Eigenpairs and Oil Companies

“Oil companies frequently use eigenvalue analysis to explore land for oil. Oil, dirt, and other substances all give rise to linear systems which have different eigenvalues, so eigenvalue analysis can give a good indication of where oil reserves are located. Oil companies place probes around a site to pick up the waves that result from a huge truck used to vibrate the ground. The waves are changed as they pass through the different substances in the ground. The analysis of these waves directs the oil companies to possible drilling sites.”

Note: This is a legacy post originally published in 2006/07/29.