The corresponding eigenvalue is the factor by which the eigenvector is scaled. Geometricallyan eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched.

If the eigenvalue is negative, the direction is reversed. However, in a one-dimensional vector space, the concept of rotation is meaningless.

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If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in Vthen v is an eigenvector of T if T v is a scalar multiple of v. This can be written as. There is a direct correspondence between n -by- n square matrices and linear transformations from an n -dimensional vector space into itself, given any basis of the vector space.

Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. If V is finite-dimensional, the above equation is equivalent to. Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations.

The prefix eigen- is adopted from the German word eigen for "proper", "characteristic". In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction.

Eigenvectors and Eigenvalues

This condition can be written as the equation. The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point.

The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right and points in the bottom half are moved to the left proportional to how far they are from the horizontal axis that goes through the middle of the painting.

The vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction.Jeff Sagarin is an American sports statistician known for his development of a method for ranking and rating sports teams in a variety of sports.

Sagarin earned a Bachelor of Science in mathematics from the Massachusetts Institute of Technology in In he moved to Bloomington, Indiana. Sagarin, like the developers of many other sports rating systemsdoes not divulge the exact methods behind his system. He offers two rating systems, each of which gives each team a certain number of points. One system, "Elo chess," is presumably based on the Elo rating system used internationally to rank chess players.

This system uses only wins and losses with no reference to the victory margin. The other system, "Predictor," takes victory margin into account. For that system the difference in two teams' rating scores is meant to predict the margin of victory for the stronger team at a neutral venue.

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For both systems teams gain higher ratings within the Sagarin system by winning games against stronger opponents, factoring in such things as home-venue advantage.

For the Predictor system, margin of victory or defeat factors in also, but a law of diminishing returns is applied. Therefore, a football team that wins a game by a margin of 7—6 is rewarded less than a team that defeats the same opponent under the same circumstances 21—7, but a team that wins a game by a margin of 35—0 receives similar ratings to a team that defeats the same opponent 70—0.

This characteristic has the effect of recognizing "comfortable" victories, while limiting the reward for running up the score. At the beginning of a season, when only a few games have been played, a Bayesian network weighted by starting rankings is used as long as there are whole groups of teams that have not played one another, but once the graph is well-connected, the weights are no longer needed.

Sagarin claims that from that point, the rankings are unbiased. Sagarin's ratings are particularly relevant in the world of American college football and basketballwhere, with hundreds of teams in NCAA Division I competition, there is no way a team can play against more than a small fraction of its competitors. Therefore, in determining the participants in championship games and tournaments, it is necessary to distinguish between teams that have compiled impressive win-loss records against strong competition and teams that have defeated weaker opponents.

In addition, sports rating systems are generally of great interest to gamblers. Gamblers use Sagarin's ratings as a source of "Power Ratings," traditionally used as a way to determine the spread between two teams.

Winston, a professor of decision sciences at Indiana UniversitySagarin advises the Dallas Mavericks about which lineups to use during games and which free agents to sign using a system called Winval.While betting on sports is only legal in a few places in the United States, such as Las Vegas, millions of office workers are involved in sports pools every week now that the football season has arrived.

Folks in the gaming business know that more than a billion dollars is wagered on every Monday Night Football game during the season. For those who wager, it may be helpful to put some science on your side when you wager, and one of the best places to do that is with the Sagarin College Football Ratings.

Jeff Sagarin

You will have to forgive the NCAA for taking titles that have been used for years and are perfectly clear, then renaming them and creating confusion in the process. If there is a way for the NCAA to assert its superior power, it does so by making everything more difficult and confusing, similar to your United States government and its IRS tax code which could reduce a sane person to tears just reading it.

Anyway, the Sagarin rating is a numerical measure of a team's strength. A hypothetical victory margin is determined by comparing the rating of the two teams after adding 2. The home edge will vary during the season.

A diminishing-returns principle exists to prevent teams from building up ratings by running up large victory margins against weak teams. Instead, it rewards teams that do well against good opponents. For Sagarin ratings and more detailed information go to: www. Following the first week of college football action, here are some facts that interested me about Sagarin's first-week ratings: 1 Washington, one of the poor to mediocre teams in the country the last several years, was rated No.

The win was the biggest upset in college football history as no AA team had ever beaten a ranked team. Michigan was ranked No. Following its horrendous loss, Michigan ended up being ranked No.

Georgia Tech was rated No. The Irish failed to score a touchdown for the first time ever in their home opener. The worst-rated A school is Florida International at No. The worst-rated AA school is the No. La Salle is a Catholic university located in Philadelphia. La Salle lost its home opener to Ursinus Ursinus is not a planet but a real liberal arts college in Pennsylvania.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. A number of areas I'm studying in my degree not a maths degree involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the explanations given in textbooks and lectures. Does anyone know of a good, fairly simple but mathematical explanation of eigenvectors and eigenvalues on the internet?

If not, could someone provide one here? In a vast number of situations, the objects you study and the stuff you can do with them relate to vectors and linear transformations, which are represented as matrices. This ranges from systems of linear equations you have to solve which occurs virtually everywhere in science and engineering to more sophisticated engineering problems finite element simulations.

It also is the foundation for a lot of quantum mechanics. It is further used to describe the typical geometric transformations you can do with vector graphics and 3D graphics in computer games.

This is a bit awkward and costly to compute in a naive fashion.

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This observation is generalized by the concept of eigenvectors. But eigenvectors only get stretched, not rotated.

The next important concept is that of an eigenbasis. This made it clearer for me: Khan Academy - Introduction to Eigenvalues and Eigenvectors I often find it easier to understand via illustration like this. Roll the pen between your palms such that when it spins, the axis of rotation matches the same vector that the pen points. Now assume we have a 3D simulation that rotates the pen in this way.

In the simulation of the rotated pen, the computer has to calculate the position of each point within the pen. The rotation is performed by a 3D transformation matrix that when multiplied by the matrix of the points in the pen, defines precisely how that pen will rotate on the 3d cartesian plane. The pencil is just a 3D matrix. There is another matrix that when multiplied, yields the correct rotation around the axis of rotation. In this little pen rolling simulation, you have the Matrix for the location of particles in the pen, and you have the matrix that says exactly how to do the 3D transform to make it rotate.

What if you wanted to know the axis of rotation of the pen, given only the pen and the transform. You would not know how to do it. Enter stage left the following equation: np. This equation asserts that if you can find a number times the transform that is the same as the dot product between the pen and the transform, that yields the axis of rotation.

You can find out which way the pen is pointing given only how the particles in the pen spin. It's a clever trick to isolate variables and discover new truths.

This methodology is useful because discovering how a Matrix times vector produces the same result as a scalar times a vector helps us find the axis of rotation that minimizes variance.By Victor Powell and Lewis Lehe.

Let's see if visualization can make these ideas more intuitive. Note three facts: First, every point on the same line as an eigenvector is an eigenvector.

Those lines are eigenspacesand each has an associated eigenvalue. Fibonacci Sequence Suppose you have some amoebas in a petri dish. Every minute, all adult amoebas produce one child amoeba, and all child amoebas grow into adults Note: this is not really how amoebas reproduce. Below, press "Forward" to step ahead a minute. The total population is the Fibonacci Sequence. Drag the circles to decide these fractions and the number starting in each state.

At this "steady state," the same number of people move in each direction, and the populations stay the same forever.

Hover over the animation to see the system go to the steady state. For more on Markov matrices, check out our explanation of Markov Chains. So far we've only looked at systems with real eigenvalues. That it can't be a complex number? For example.

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You'll see that whenever the eigenvalues have an imaginary part, the system spirals, no matter where you start things off. Learning more We've really only scratched the surface of what linear algebra is all about. Eigenvectors and Eigenvalues Explained Visually.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up.

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I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to understand. I found that there are many application of eigenvalue and eigenvector in multivariate analysis. For example. I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically.

I think my former understanding may be too naive so that I cannot find the link between eigenvalue and its application in principal components and others. I know how to induce almost every step form the assumption to the result mathematically. I'd like to know how to intuitively or geometrically understand eigenvalue and eigenvector in the context of multivariate analysis in linear algebra is also good. Personally, I feel that intuition isn't something which is easily explained. Intuition in mathematics is synonymous with experience and you gain intuition by working numerous examples.

With my disclaimer out of the way, let me try to present a very informal way of looking at eigenvalues and eigenvectors. First, let us forget about principal component analysis for a little bit and ask ourselves exactly what eigenvectors and eigenvalues are. A typical introduction to spectral theory presents eigenvectors as vectors which are fixed in direction under a given linear transformation.

The scaling factor of these eigenvectors is then called the eigenvalue. Under such a definition, I imagine that many students regard this as a minor curiosity, convince themselves that it must be a useful concept and then move on. It is not immediately clear, at least to me, why this should serve as such a central subject in linear algebra. Eigenpairs are a lot like the roots of a polynomial. It is difficult to describe why the concept of a root is useful, not because there are few applications but because there are too many.

If you tell me all the roots of a polynomial, then mentally I have an image of how the polynomial must look. For example, all monic cubics with three real roots look more or less the same. So one of the most central facts about the roots of a polynomial is that they ground the polynomial. A root literally roots the polynomial, limiting it's shape.

Eigenvectors are much the same. If you have a line or plane which is invariant then there is only so much you can do to the surrounding space without breaking the limitations. So in a sense eigenvectors are not important because they themselves are fixed but rather they limit the behavior of the linear transformation. Each eigenvector is like a skewer which helps to hold the linear transformation into place. Very very, very roughly then, the eigenvalues of a linear mapping is a measure of the distortion induced by the transformation and the eigenvectors tell you about how the distortion is oriented.

It is precisely this rough picture which makes PCA very useful. If this ellipsoid was very flat in some direction, then in a sense we can recover much of the information that we want even if we ignore the thickness of the ellipse. This what PCA aims to do. The eigenvectors tell you about how the ellipse is oriented and the eigenvalues tell you where the ellipse is distorted where it's flat.

If you choose to ignore the "thickness" of the ellipse then you are effectively compressing the eigenvector in that direction; you are projecting the ellipsoid into the most optimal direction to look at.Since the early s, Jeff Sagarin has been publishing sports team ratings. For most sports, including the NFL, his ratings are calculated so that the difference between two opponent's ratings, plus a home field adjustment, forecast a game's point spread.

His ratings are widely recognized as some of the best around.

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Sagarin has never published his exact algorithms, but we can easily build a very good facsimile. Excel has a powerful tool called "Solver. In fact, you do don't even see it on the Tools menu until you enable it from the "Tools Add Ins If you go to Microsoft's on-line help site for Solver, the example problem provided is an exercise estimating point spreads for NFL games.

The sample spreadsheet is for all the game scores from the season. Basically all you do is create a table of ratings for each team. The ratings don't have to mean anything yet. For now they can be your best guess, or all ones, or anything.

Solver will calculate them later. Then for each game, you calculate what the ratings suggest should be the point spread. The ratings are intended to work just like Jeff Sagarin's ratings.


If team A's rating is 5 and team B's rating is 8, then when team A plays team B the point spread should be 3 in favor of team B. Factoring in a league-wide value for home field advantage, say 3, and the spread becomes 6 if team B is at home. Next, using the LOOKUP function to grab the ratings from the table, you calculate the error between the expected spread and the actual result for each game.

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Square the error as every good statistician would. In a cell, sum all the squared errors for all the games in the season. In another cell, enter a point value for home field advantage points is a good initial guess. Solver takes over from here. In the Solver dialog box, you tell it to minimize the value in the cell for the sum of squared errors. Then you tell it to do so by varying the values in the table for the team ratings and the cell for the home field advantage.

You can also add in a constraint that says the average for all the teams' ratings should be zero, so that good teams will have positive ratings and poor teams will have negative ratings. Solver will compute the team ratings necessary to best fit the actual point spreads. And now you have your very own homemade Sagarin ratings. I noticed that Sagarin's average rating is 20 instead of 0, which makes sense because the average NFL score is about 20 points.

So I altered the Solver constraint accordingly. For the season, including the playoffs, the homemade ratings were nearly identical to Sagarin's. Solver uses a "brute force" numerical iteration method, and Sagarin's method is unknown.

Sagarin may also weight recent games heavier. Notice how the difference in the Giants' rating is one of the more significant. The Giants finished the season on quite a win streak.

Doug Drinen of Pro-Football-Reference. His post includes a good discussion on the advantages and disadvantages of a pure margin of victory ranking system.

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