R help - Aug 2006 - Geometrical Interpretation of Eigen value and Eigen vector

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

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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]]
>
> ______________________________________________
> R-help at stat.math.ethz.ch 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

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 <arun.kumar.saha at gmail.com>
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]]
>
> ______________________________________________
> R-help at stat.math.ethz.ch 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.
>

Arun,

have a look at:

http://149.170.199.144/multivar/eigen.htm

HTH,
Dirk

"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?