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What the Heck are Eigenvalues and Eigenvectors?

Updated on June 2, 2010
Google campus
Google campus

How to Quickly End a Dinner Conversation

There are certain words that will put a speedy end to any dinner conversation.  Eigenvalue, eigenvector, and eigenfunction are probably in the top 100 such words. They are used in linear algebra and unfortunately, when they are used, they are rarely explained.  If you do happen to find them, then most likely, you have opened a technical book by mistake.

Consider the following conversation as an example:

Guest:  Do you feel that Google will continue its dominance of search?

Mathematician:  I feel confident that Google's use of eigenvectors places it in a unique position of importance among...

Guest:  (yawn) 

Last year, a book was written which attempted to explain the mathematics behind Google's page rank algorithm.  One of the chapters of the book is called:  The $25 billion dollar Eigenvector.

My goal in this hub is to explain the intuitions behind these terms in an effort to explain what they are, how they are used, and some basic ideas about them.

Eigen this and Eigen that

The prefix "eigen" is itself a German word which means "proper" or "characteristic (see here). Unfortunately, this doesn't help us very much in understanding what they are other than to suggest that they were invented by a German mathematician. There is some truth to this since possibly the first person to give them their current name was the German mathematician David Hilbert (see, here). Although, it may have also been the German physicist Hermann Ludwig Ferdinand von Helmholtz who was first (see, here).

Initially, eigenvalues were called "Proper Values" in the United States but that term is no longer used. Today, they are universally called eigenvalues and eigenvectors (for a complete history of the term, see here).

Eigenvalues and Eigenvectors defined

An eigenvalue is a number that is derived from a square matrix.  A square matrix is itself just a collection of n rows of n numbers.  An eigenvalue is usually represented by the Greek letter lamdba (λ).

Let A be a square matrix (a collections of n rows of n numbers which means that there are n x n numbers in total).

Let x be a nonzero vector.  A vector is just a column of numbers.  A nonzero vector is any vector where not all the numbers are 0.   By convention, a vector that consists entirely of 0's is called the 0 vector.

We say that a number is the eigenvalue for this square matrix if and only if there exists a nonzero vector x such that Ax = λx where:

A is the square matrix

x is the nonzero vector

λ is a nonzero value.

In this circumstance, λ is the eigenvalue and x is the eigenvector.

So, who cares?

So far, we've shown that certain square matrices satisfy an equation such that:

Ax = x

So what?  Why should I care about matrices, nonzero vectors, eigenvalues, and eigenvectors?

The major reason for studying eigenvalues and eigenvectors is that they are used in many important mathematical results.  Perhaps, it makes sense to show one example of a real world application before answering the other questions.

The Tacoma Narrows Bridge today
The Tacoma Narrows Bridge today

Eigenvalues and the Collapse of the Tacoma Narrows Bridge

On July 1, 1940, the Tacoma Narrows Bridge opened in Washington state.  It connected the city of Tacoma with the Kitsap Penninsula and ran over the Tacoma Narrows which is a strait across the Puget Sound.  Four months after it was built, it collapsed.  This was captured in film and was later nicknamed "Gallopin' Gertie".  The full story can be found here.

Believe it or not but this collapse can be explained in engineering terms using the idea eigenvalues.

The Collapse of the Tacoma Narrows Bridge in 1940

The Bridge Collapse and Eigenvalues

Why did the bridge collapse?

One explanation centers around natural frequencies.  The natural frequency is "the frequency at which a system naturally vibrates once it has been set into motion" (from this article).  In other words, the natural frequency is the characteristic motion of structure.  It's the motion that a structure takes on in response to wind or being walked on.  It is especially important in the design of musical instruments and in the tuning of radios.

Mathematically, the natural frequency can be characterized by the eigenvalue of the smallest magnitude.

The model suggests that the "oscillations of the bridge were caused by the frequency of the wind being too close to the natural frequency of the bridge." (from this article)  When frequencies match, they compound which proved too strong a force for the bridge.

The same type of collapse can happen with soldiers marching.  If soldiers march in lock step too close to the natural frequency of a bridge, then it is possible, under some circumstances, for the bridge to collapse.


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    • profile image

      olivier 2 years ago

      Please go ahead with explainations..You did not finish ? Where are paragraphs that follow '... for the bridge to collapse' ?

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      einstein baby 4 years ago

      Actually the correct explanation is that the bridge fell due to implosion of the particles caused by excessively released dark enery as a result of wind energy bending the space-time of the vibration of the bridge. This explanation is not found in any of the text books of the universe.

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      Erwin 4 years ago

      This article says nothing about practical usage of Eigen values and Eigen vectors. Just a waste of time.

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      om 4 years ago

      stop fighting to each others , this is not a betel field

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      Erik 4 years ago

      Sorry, but the collapse of the Tacoma Narrows Bridge's is a really bad example for a resonance catastrophe, as suggested here, because it really isn't -- therefore it is also not analogous to marching soldiers, whose unison gait excite the characteristic frequency of the bridge.

      The collapse of the T N bridge is a nonlinear phenomenon, and probably was caused by aerodynamic flutter (see for instance here,

      Also, the explanation that the 'wind has the same frequency as the bridge' makes absolutely no sense, considering that the wind as such most likely did not have any oscillatory motion by itself.

      I think it's great to popularize such topics, but it'd be good to do a little research before bringing about such explanations. Alas, perhaps I can't blame the writer for this mistake, as the false explanation for the collapse of the TN bridge is extremely widely spread, even in text books.

    • Manna in the wild profile image

      Manna in the wild 4 years ago from Australia

      The fundamental frequency is the lowest component in a non sinusoidal waveform.

    • profile image

      manna in the wild 4 years ago

      Why smallest?

      Mathematically, the. "natural frequency can be characterized by the eigenvalue of the. smallest magnitude.

    • profile image

      eigenCheers 5 years ago

      This following link is much better than this article where you can find meat in just firt 2-3 slides :)

    • profile image

      Tliimfee 5 years ago

      So way can eigenvectors and eigenvalues recreate data? What is the intuition behind that? What does it mean to be the principal eigenvalue, how does that capture most of the data?

    • profile image

      sham 5 years ago

      where is the meat?????

    • profile image

      Joshi 5 years ago

      @ anothermathgeek, You are a rare breed my friend- You can bring the beauty of math to the masses. Keep up the good work. I'll be following your posts from now.

    • mathsciguy profile image

      mathsciguy 6 years ago from Here, there, and everywhere

      I remember a funny story about eigenvalues that your intro reminded me of. So, our department head was talking about Markov Chains and brought up the theorem about any Markov chain having 1 as an eigenvalue. He showed us a proof of it and then said, "Now, if you are ever driving in your car and someone stops you and shows a Markov chain matrix and asks you for an eigenvalue of it, you will be able to know at least one." Loved that guy.

    • anothermathgeek profile image

      anothermathgeek 6 years ago from East Bay, California

      Hi Aaron,

      This is meant as a high level overview. Based on the feedback I've received, I'll write another hub that goes a bit deeper into the notation and the mathematics.

    • profile image

      Aaron 6 years ago

      That's it??? Seems like a nice introduction, but where is the substance?

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      ssss 7 years ago

      can you tell what is the eigen value response between the two square matrix

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      l2oss 7 years ago

      pretty sweet dude. One thing about the article isn't totally accurate though, as eigenvalues CAN be zero. I like the way you explain things it is very clear and concise.

    • profile image

      Pramod Rawat 7 years ago

      Like the hub.

      Thanks to write this please write more.

    • ss sneh profile image

      ss sneh 7 years ago from the Incredible India!

      Hi! All particles in the universe have tendency to vibrate.

      It's the same with a bridge...the particles that constitute that bridge also vibrate with different frequencies- called natural frequencies.

      When wind blows on a bridge it is possible that the wind can make some of these particles to vibrate with the same frequencies. When vibrations of two particles fall constructively then the amplitude of the combined vibrations of those particles become maximum.

      If you give continuous external excitation with a certain frequency... like continuous blowing of wind with a particular rhythm- then that can cause all particles to vibrate with a combined frequency with the maximum amplitude - called resonance.

      This concentration of energy which can cause maximum amplitude - maximum displacement from the horizontal position of that bridge - can damage it if the resonance effect is ignored while designing.

      I wonder what the phenomenon of resonance has got with Google ranking?-- Thanks

    • anothermathgeek profile image

      anothermathgeek 7 years ago from East Bay, California

      Thanks, Larry. I'm planning to write more as I have time.

    • Ben Evans profile image

      Ben Evans 7 years ago

      Good explanation of eigenvalues. It is a very interesting seeing that it is part of Google's algorithm.



    • nicomp profile image

      nicomp really 7 years ago from Ohio, USA

      Wonderful. You are a rare bird; a geek who can explain stuff. Thank you!

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      Dan 7 years ago

      Thanks - this article is the best I've found on the _whole_ internet to discuss/explain what eigenvalues/functions really are. Please continue the article !

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      botsbry 8 years ago

      Does this finish with the paragraph about soldiers or is there more?

    • anothermathgeek profile image

      anothermathgeek 8 years ago from East Bay, California

      Hi Subha,

      Click on the link above to read how the eigenvalue method is used by Google.


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      subha 8 years ago

      how the eigenvalue method used in search engine?