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What are eigen values?

physlink.com

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Re: What are eigen values?

#2

Post by baddox »

After reading the article, I have but one question: What are eigen values?

A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usefulness of definite integrals. In those cases, you end up calculating the area under the curve, but the curve happens to be the graph of a constant function or a first-order polynomial. Calculating such areas can easily be done by using the elementary formula for the area of a trapezoid.

Re: What are eigen values?

#3

Post by dtf »

baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

Here's another intuitive introduction in this article explaining the SVD:

http://www.ams.org/featurecolumn/archive/svd.html

Re: What are eigen values?

#4

Post by swombat »

This is a pretty good explanation for an essentially mathematical/abstract concept.

Another name for eigenvectors is "identity" vectors, iirc (it's been a few years...).

Re: What are eigen values?

#5

Post by antiform »

baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

In linear algebra, for a given linear transformation (a certain kind of matrix which generally represents some operation) M, an eigenvalue of M is a scalar c such that given a non-zero vector v (called the eigenvector), Mv = cv, where multiplication on the left is matrix multiplication and multiplication on the right is multiplication of v by the scalar c.

The definition is significant because it says that for a certain vector v, the transformation M, no matter how complicated it may be, just scales v by a factor of c. This is useful, for instance, if you want to determine an axis by which to evaluate the range of the transformation, because by choosing an eigenvector, you are choosing a "simple" or "natural" perspective from which to evaluate the range.

Re: What are eigen values?

#6

Post by tybris »

dtf wrote: post #3
baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

Here's another intuitive introduction in this article explaining the SVD: http://www.ams.org/featurecolumn/archive/svd.html

Thanks, that one's really good.

Re: What are eigen values?

#7

Post by gjm11 »

swombat wrote: post #4

This is a pretty good explanation for an essentially mathematical/abstract concept. Another name for eigenvectors is "identity" vectors, iirc (it's been a few years...).

I have never heard eigenvectors called identity vectors, and a quick Google search turns up no examples of that usage.

Re: What are eigen values?

#8

Post by jharsman »

baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

If you're doing graphics or physics, calculating the eigen vector for the eigen value 1 gives you the axis of rotation for a rotation matrix.

That's a pretty straight forward application that's also obvious from the definition of eigen values and eigen vectors.

Re: What are eigen values?

#9

Post by cracki »

swombat wrote: post #4

This is a pretty good explanation for an essentially mathematical/abstract concept. Another name for eigenvectors is "identity" vectors, iirc (it's been a few years...).

"identity vectors"... do you perhaps mean "unit vectors", of length 1?
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