MAT225 Section Summary: 5.1

Eigenvalues and Eigenvectors

Summary

We're considering the transformation tex2html_wrap_inline293 . Eigenvectors provide the ideal basis for tex2html_wrap_inline295 when considering this transformation. Their images under the transformation are simple scalings.

Eigenstuff: An eigenvector of tex2html_wrap_inline297 is a nonzero vector tex2html_wrap_inline299 such that tex2html_wrap_inline301 . The scalar tex2html_wrap_inline303 is called the eigenvalue of A corresponding to tex2html_wrap_inline299 . There may be several eigenvectors corresponding to a given tex2html_wrap_inline303 .

The idea is that an eigenvector is simply scaled by the transformation, so the actions of a transformation are easily understood for eigenvectors. If we could write a vector as a linear combination of eigenvectors, then it would be easy to calculate its image: if there are n eigenvectors tex2html_wrap_inline313 , with n eigenvalues tex2html_wrap_inline317 , then if

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then

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Nice, no?

If tex2html_wrap_inline303 is an eigenvalue of matrix A corresponding to eigenvector tex2html_wrap_inline323 , then

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This means the

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which is equivalent to

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So tex2html_wrap_inline323 is in the null space of tex2html_wrap_inline327 . If the null space is trivial, then tex2html_wrap_inline323 is the zero vector, and tex2html_wrap_inline303 is not an eigenvalue. Alternatively, all vectors in the null space are eigenvectors corresponding to the eigenvalue tex2html_wrap_inline303 .

As for determining the eigenvectors and eigenvalues, there is some cases in which this is extremely easy:

The eigenvalues of a diagonal matrix are the entries on its diagonal. More generally,

Theorem 1: The eigenvalues of a triangular matrix are the entries on its main diagonal.

Theorem 2: If tex2html_wrap_inline335 are eigenvectors corresponding to distince eigenvalues tex2html_wrap_inline337 of an tex2html_wrap_inline339 matrix A, then the set tex2html_wrap_inline343 is linearly independent.

The eigenvectors and difference equations portion of this section can be illustrated with the example of the Fibonacci numbers transformation: recall that the Fibonacci numbers are those obtained by the recurrence relation

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and tex2html_wrap_inline345 and tex2html_wrap_inline347 .

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where

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The eigenvalues of this matrix are approximately tex2html_wrap_inline349 and -0.618033988749894. tex2html_wrap_inline351 is the so-called ``golden mean'', which is a nearly sacred number in nature, well approximated by the ratio of consecutive Fibonacci numbers.

An eigenvector corresponding to the golden mean (normalized to have a norm of 1) is approximately

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so that

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LONG ANDREW E
Sat Jan 29 20:58:20 EST 2011