[R] Geometrical Interpretation of Eigen value and Eigen vector

Izmirlian, Grant (NIH/NCI) [E] izmirlig at mail.nih.gov
Mon Aug 14 04:39:22 CEST 2006

Ok, I had a look at it. It seems like awefully far to dig for the main point which is easily 
summarized in a few sentences.

If we super-impose the pre-image and image spaces (plot the input and output in the same
picture), then in 1 dimension, a linear function, say 'a x', takes its input, x, and stretches
it by a factor |a|. If 'a' is negative, then the direction that 'x' points is reversed.

Understanding several dimensions, as is usually the case, requires us to refine our
understanding of the 1-dimensional case.  In several dimensions, a linear function, 
say 'A x' (where 'A' is an m by m matrix and 'x' is an 'm' vector) will result in the stretching
of the input, 'x', along the direction its pointing, by a factor 'a'. However, this is the case
_only_ if 'x' lies in one of the 'characteristic directions' corresponding to 'A'. Since 'A'
is an m by m matrix, there will be at most m such 'characteristic directions'.  Each of the
characteristic directions has its associated stretching factor.  The characteristic directions
are called eigenvectors and the corresponding stretching factors are called eigenvalues.

Think about what this means in 1-dimension (hint: there's only one dimension so only
one possible direction).

The number of linearly independent characteristic directions (eigenvectors) is called the
rank of the matrix, A.  If you understand the concept of 'basis' then you know that any
m vector can be expressed in terms of the basis of eigenvectors of 'A' (that is unless A is not
of 'full rank' and has less than m linearly independent eigenvectors, in which case we decomponse
'x' into two orthogonal components, one as a linear combination of the eigenvectors of A and the other
gets mapped to 0 by A.)

Thus to each input 'x' is assigned an output 'y' which is the sum of coefficients in the eigenvector
basis representation of 'x' times corresponding eigenvalues.  This can be understood as the  
diagonalization of 'A'.  By the way, the referenced page was in error because the singular value
decomposition (I think the page actually called it the single value decomposition...free translation(s).com 
anyone) is not the same thing as the diagonalization.

There, it took a little more than a few sentences, but at least by the close of the second paragraph
one gets the basic idea.

Now, in closing, Arun, please spend some time thinking about the answer to your question before
you cut and paste it into your homework assignment.

-----Original Message-----
From: Dirk Enzmann [mailto:dirk.enzmann at uni-hamburg.de]
Sent: Sat 8/12/2006 7:01 AM
To: r-help at stat.math.ethz.ch
Cc: arun.kumar.saha at gmail.com
Subject: Re: [R] Geometrical Interpretation of Eigen value and Eigen vector

have a look at:


"Arun Kumar Saha" <arun.kumar.saha at gmail.com> wrote:

> It is not a R related problem rather than statistical/mathematical. However
> I am posting this query hoping that anyone can help me on this matter. My
> problem is to get the Geometrical Interpretation of Eigen value and Eigen
> vector of any square matrix. Can anyone give me a light on it?

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