Eigenvalues/Eigenvectors Real World Application
Eigenvalues and Eigenvectors are important to engineers because they basically show what the the matrix is doing. Eigenvalues/vectors are used by many types of engineers for many types of projects. Some of those applications include noise reduction in cars, stereo systems, vibration analysis, material analysis, and structural analysis. The application we will be looking at is structural analysis and in particular the 1940 Tacoma Narrows bridge collapse. As a mechanical engineer a large portion of many jobs is analyzing how forces interact with structures. Using Eigenvalues and Eigenvectors engineers can prevent disasters like the Tacoma bridge.
The video above shows how Eigenvectors relate to the matrix. The blue arrows are the Eigenvectors. It is important to note that the blue arrows do not change direction. These are the Eigenvectors of the matrix. The other colored arrows do change direction when the matrix is manipulated, therefore they are not Eigenvectors. Also the photo below is an example of an Eigenvector. In these pictures the Mona Lisa image has been sheared to the right. The blue arrow is a Eigenvector of this "shear-map" because it does not change direction. Although it looks like both axis have been manipulated if you look at Mona lisa's eyes you can see that in reality they stay level.