Given a matrix A, what do eigenvectors and eigenvalues of A imply? I know how to calculate them but I want to understand WHY do we need to find them? In what application this is important
This page may help graphically some of the applications: http://setosa.io/ev/eigenvectors-and-eigenvalues/
All music is just eigenvalues and eigenvectors. The strings of a guitar, a sitar or a santoor - they resonate at their eigenvalue frequencies. The membranes of percussion instruments like the Indian tabla, drums, etc. resonate at their eigenvalues and move according to the two dimensional eigenvectors.
Eigenvectors of your data set matrix correspond to directions of maximum variance, ordered in decreasing marginal increase in variance by decreasing corresponding eigenvalues. This is the main idea behind principal component analysis (PCA), a dimensionality reduction trick often used in machine learning and AI.
Eigenvalues of the system matrix of a linear system tell you information about the stability and response of your system. For a continuous system, the system is stable if all eigenvalues have negative real part (located in the left half complex plane). For a discrete system, the system is stable if all eigenvalues have magnitude less than 1 (inside the unit circle in the complex plane)
Eigenvalues of matrices associated with graphs, like the adjacency matrix and the Laplacian matrix. They relate to various structural properties of the graph. For instance, the number of 0 eigenvalues of the Laplacian matrix is equal to the number of components of the graph. The number of distinct eigenvalues of the adjacency matrix is a lower bound for one plus the diameter of the graph (the size of the longest path of the graph), and so on.
The eigenvalues and eigenvectors of a matrix are often used in the analysis of financial data and are integral in extracting useful information from the raw data. They can be used for predicting stock prices and analyzing correlations between various stocks, corresponding to different companies. They can be used for analyzing risks. There is a branch of Mathematics, known as Random Matrix Theory, which deals with properties of eigenvalues and eigenvectors, that has extensive applications in Finance, Risk Management, Meteorological studies, Nuclear Physics, etc.
Google Search Results
The searching algorithm of Google diagonalizes a giant matrix and with the SVD (Singular value decomposition) method, and according to the eigenvalue assigned to every website, it shows you the best results. See the paper about this here: http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf
Eigenvalues are possible measurement results of an observable represented by an operator.
Eigenvectors of the moment of inertia tensor represent the "main axes" around which a solid body can stably rotate, and the corresponding eigenvalues are scalar moments of inertia along those axes.
These summaries pulled from : https://www.quora.com/What-are-some-very-good-and-practical-uses-of-eigenvalues-of-a-matrix