Geometrical Interpretation of Eigen value and Eigen vector

5 messages
Open this post in threaded view
|

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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
Open this post in threaded view
|

Re: Geometrical Interpretation of Eigen value and Eigen vector

 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 ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
Open this post in threaded view
|

Re: Geometrical Interpretation of Eigen value and Eigen vector

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