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What the Heck are Eigenvalues and Eigenvectors?

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.

Comments

olivier on April 08, 2015:

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

einstein baby on July 14, 2013:

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.

Erwin on May 04, 2013:

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

om on March 24, 2013:

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

Erik on March 04, 2013:

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, http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge...

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 from Australia on January 24, 2013:

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

manna in the wild on January 24, 2013:

Why smallest?

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

eigenCheers on November 15, 2012:

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

http://www.scribd.com/doc/23290356/Eigenvalues-and...

Tliimfee on October 11, 2012:

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?

sham on April 19, 2012:

where is the meat?????

Joshi on March 11, 2012:

@ 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 from Here, there, and everywhere on April 09, 2011:

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 (author) from East Bay, California on March 15, 2011:

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.

Aaron on March 15, 2011:

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

ssss on August 28, 2010:

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

l2oss on August 26, 2010:

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.

Pramod Rawat on July 17, 2010:

Like the hub.

Thanks to write this please write more.

ss sneh from the Incredible India! on July 08, 2010:

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 (author) from East Bay, California on June 23, 2010:

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

Ben Evans on May 31, 2010:

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

Cheers,

Bem

nicomp really from Ohio, USA on April 10, 2010:

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

Dan on April 09, 2010:

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 !

botsbry on January 02, 2010:

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

anothermathgeek (author) from East Bay, California on September 09, 2009:

Hi Subha,

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

-Larry

subha on September 09, 2009:

how the eigenvalue method used in search engine?

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