Why eigenvalues and eigenvectors

And, accordingly, you can identify the image as the tiger. The solution to real-world problems often depends upon processing large volume of data representing different variables or dimensions. For example, take the problem of predicting the stock prices. Here the dependent value is stock price and there are a large number of independent variables on which the stock price depends. Using large number of independent variables also called features , training one or more machine learning models for predicting the stock price will be computationally intensive.

Such models turn out to be complex models. This will result in simpler and computationally efficient models. This is where eigenvalues and eigenvectors comes into picture. Feature extraction algorithms such as Principal component analysis PCA depend on the concepts of Eigenvalues and Eigenvectors to reduce the dimensionality of data features or compress the data data compression in form of principal components while retaining most of the original information.

In PCA, the eigenvalues and eigenvectors of features covariance matrix are found and further processed to determine top k eigenvectors based on the corresponding eigenvalues. Thereafter, the projection matrix are created from these eigenvectors which are further used to transform the original features into another feature subspace.

With smaller set of features, one or more computationally efficient models can be trained with the reduced generalization error. Thus, it can be said that Eigenvalues and Eigenvectors concepts are key to training computationally efficient and high performing machine learning models.

D ata scientists must understand these concepts very well. Finding Eigenvalues and Eigenvectors of a matrix can be useful for solving problems in several fields such as some of the following wherever there is a need for transforming large volume of multi-dimensional data into another subspace comprising of smaller dimensions while retaining most information stored in original data. The primary goal is to achieve optimal computational efficiency.

Eigenvectors are the vectors which when multiplied by a matrix linear combination or transformation results in another vector having same direction but scaled hence scaler multiple in forward or reverse direction by a magnitude of the scaler multiple which can be termed as Eigenvalue. In simpler words, eigenvalue can be seen as the scaling factor for eigenvectors. Here is the formula for what is called eigenequation.

Note that the new vector Ax has different direction than vector x. Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row.

Whenever there is a complex system having large number of dimensions with a large number of data, eigenvectors and eigenvalues concepts help in transforming the data in a set of most important dimensions principal components.

This will result in processing the data in a faster manner. Your email address will not be published. Time limit is exhausted. Thank you for visiting our site today. Self adjoint operators have the following two key properties that allows them to make sense as measurements as a consequence of infinite dimensional generalizations of the spectral theorem :. As a more concrete and super important example, we can take the explicit solution of the Schrodinger equation for the hydrogen atom.

In that case, the eigenvalues of the energy operator are proportional to spherical harmonics :. The energy difference between two energy levels matches experimental observations of the hydrogen spectral series and is one of the great triumphs of the Schrodinger equation.

The general Schrodinger equation can be simplified by separation of variables to the time independent Schrodinger equation , without any loss of generality:.

And since E is a constant the energy , this is just an eigenvalue equation. Have a look at: Real world application of Fourier series to get a feeling for separation of variables works for a simpler equation like the heat equation. Heuristic argument of why Google PageRank comes down to a diagonalization problem. Therefore, one can feel that theoretically, an "iterative approach" cannot work: we need to somehow solve the entire system in one go.

And one may hope, that once we assign the correct importance to all nodes, and if the transition probabilities are linear, an equilibrium may be reached:. The equilibrium also happens on the vector with eigenvalue 1, and convergence speed is dominated by the ratio of the two largest eigenvalues. I would like to direct you to an answer that I posted here: Importance of eigenvalues.

I feel it is a nice example to motivate students who ask this question, in fact I wish it were asked more often. Personally, I hold such students in very high regard. These are important invariants of linear transformations.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. Asked 10 years, 8 months ago. Active 1 year, 3 months ago. Viewed k times. Ryan Ryan 4, 6 6 gold badges 17 17 silver badges 10 10 bronze badges. It offers a pretty complete answer to the question.

I am extremely surprised this question hasn't already come up. Show 3 more comments. Active Oldest Votes. Slightly Longer Answer There are a lot of problems that can be modeled with linear transformations, and the eigenvectors give very simply solutions. Sanchit 3 2 2 bronze badges. Arturo Magidin Arturo Magidin k 49 49 gold badges silver badges bronze badges. Chapeau bas! Add a comment.

Tanner 1. I would like just to say that this short explanation was great! I find this good simple example very precious to serve as a motivation for eigenvalues, matrizes, etc. Thank you! Why bother? Show 1 more comment. For example, it could make the student naively ask, "why does the basis matter at all?

It would be nice to be able to address this without assuming they already know a lot of linear algebra. As the existence of a Jordan Block signals that some transformations act on certain combinations of axes that are inherently non-decomposable. Sridhar Thiagarajan 2 2 gold badges 5 5 silver badges 20 20 bronze badges. Herb Herb 2 2 silver badges 5 5 bronze badges. In general the method of characteristics for partial differential equations can be had for arbitrary first-order quasilinear scalar PDEs defined on any smooth manifold.

SChepurin SChepurin 6 6 silver badges 8 8 bronze badges. Why it is bad? Intuitively, there exist some strong relation between two such Matrices. Now Eigen Values are a necessary condition to check so but not sufficient though!

Let make my statement clear. Srijit Srijit 4 4 silver badges 11 11 bronze badges.