# Re: [R] Geometrical Interpretation of Eigen value and Eigen vector

From: Gabor Grothendieck <ggrothendieck_at_gmail.com>
Date: Thu 10 Aug 2006 - 23:18:08 EST

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@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]]
>
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