[R] Matrix question

[apjaworski@mmm.com]

Gideon,

Eigenvectors are normalized to unit length.  The first eigenvector
calculated by R is equal (ignoring the signs of course) to your stable
distribution vector divided by its length.

Andy

__________________________________
Andy Jaworski
518-1-01
Process Laboratory
3M Corporate Research Laboratory
-----
E-mail: apjaworski at mmm.com
Tel:  (651) 733-6092
Fax:  (651) 736-3122


|---------+-------------------------------->
|         |           GIDEON WASSERBERG    |
|         |           <wasserberg at wisc.edu>|
|         |           Sent by:             |
|         |           r-help-bounces at stat.m|
|         |           ath.ethz.ch          |
|         |                                |
|         |                                |
|         |           04/13/2004 18:28     |
|---------+-------------------------------->
  >-----------------------------------------------------------------------------------------------------------------------------|
  |                                                                                                                             |
  |      To:       "R-help at lists.R-project.org" <R-help at stat.math.ethz.ch>                                                      |
  |      cc:                                                                                                                    |
  |      Subject:  [R] Matrix question                                                                                          |
  >-----------------------------------------------------------------------------------------------------------------------------|




Dear Friends

I am doing a simple matrix analysis to calculate the eigenvalue,
eigenvector using R for the below matrix, and comparing the result to those
obtained from a projection (using excel)

THE MATRIX:
> c
     [,1] [,2] [,3]
[1,]  0.0  2.0    2
[2,]  0.8  0.0    0
[3,]  0.0  0.8    0


The dominant eigenvalue comes out comparable to that calculated
numerically, but the eigenvectors do not( see below)!


EIGENVALUES (calculated by R):

> eigen(c)
$values
[1]  1.5564082+0.000000i -0.7782041+0.465623i -0.7782041-0.465623i

EIGENVALUE numerically calculated: 1.556408145


EIGENVECTORS (calculated by R):
$vectors
              [,1]                  [,2]                  [,3]
[1,] -0.8658084+0i  0.6476861+0.0000000i  0.6476861+0.0000000i
[2,] -0.4450290+0i -0.4902997-0.2933611i -0.4902997+0.2933611i
[3,] -0.2287467+0i  0.2382837+0.4441499i  0.2382837-0.4441499i

Stable age distribution (calculated numerically):

0.562365145
0.289057934
0.148576921


My questions are:
1. Both eigenvalue and eigenvectors are associated with some imaginary
value (i). How should I relate to that information? 2. More importantly, a.
I presume the 1st eigenvector collumn [,1] should correspond to the
dominant eigenvalue. How come then that it comes out different from the one
calculated numerically? Is there some conversion I should do?

Many thanks

Gideon


Gideon Wasserberg (Ph.D.)
Wildlife research unit,
Department of wildlife ecology,
University of Wisconsin
218 Russell labs, 1630 Linden dr.,
Madison, Wisconsin 53706, USA.
Tel.:608 265 2130, Fax: 608 262 6099

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