Geometrical Interpretation of Eigen value and Eigen vector

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Geometrical Interpretation of Eigen value and Eigen vector

Arun.stat
Dear all,

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?

Thanks and regards,
Arun

        [[alternative HTML version deleted]]

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Re: Geometrical Interpretation of Eigen value and Eigen vector

Simon Wood-4
You can decompose a symmetric matrix A as
A=UDU'
where U is a matrix of eigenvectors (in its columns), and D is a diagonal
matrix of eigenvalues. Since A is symmetric, U is orthogonal. So what A does
to a vector x when you form Ax has a simple geometerical interpretation:
1. x is rotated into the `eigenspace' of A, by U'
2. the elements of the rotated x are rescaled by multiplication by the  
eigenvalues  of A.
3. The reverse of the rotation from step 1 is applied to the rescaled rotated
x, by U.    

Any use?

> Dear all,
>
> 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?
>
> Thanks and regards,
> Arun
>
> [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html and provide commented, minimal,
> self-contained, reproducible code.

--
> Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
> +44 1225 386603  www.maths.bath.ac.uk/~sw283

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Re: Geometrical Interpretation of Eigen value and Eigen vector

Gabor Grothendieck
In reply to this post by Arun.stat
A matrix M can be thought of as a linear transformation which maps
input vector x to output vector y:

     y = Mx

The eigenvectors are those "directions" that this mapping preserves.
That is if x is an eigenvector then y = ax for some scalar a.  i.e.
y lies in the same one dimensional space as x.  The only difference
is that y is dilated or contracted and possibly reversed and the scale factor
defining this dilation/contraction/reversal which corresponds to a particular
eigenvector x is its eigenvalue:  i.e. y = ax (where a is a scalar,
the eigenvalue, corresponding to eigenvector x).

In matrix terms, the eigenvectors form that basis in which the
linear transformation M has a diagonal matrix and the diagonal
values are the eigenvalues.

On 8/10/06, Arun Kumar Saha <[hidden email]> wrote:

> Dear all,
>
> 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?
>
> Thanks and regards,
> Arun
>
>        [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

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Re: Geometrical Interpretation of Eigen value and Eigen vector

Dirk Enzmann
In reply to this post by Arun.stat
Arun,

have a look at:

http://149.170.199.144/multivar/eigen.htm

HTH,
Dirk

"Arun Kumar Saha" <[hidden email]> 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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https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Geometrical Interpretation of Eigen value and Eigen vector

izmirlig
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:[hidden email]]
Sent: Sat 8/12/2006 7:01 AM
To: [hidden email]
Cc: [hidden email]
Subject: Re: [R] Geometrical Interpretation of Eigen value and Eigen vector
 
Arun,

have a look at:

http://149.170.199.144/multivar/eigen.htm

HTH,
Dirk

"Arun Kumar Saha" <[hidden email]> 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?

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.