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st: manipulation of eigenvalues


From   "Katsuhide Isa" <katu@ier.hit-u.ac.jp>
To   Statalist <statalist@hsphsun2.harvard.edu>
Subject   st: manipulation of eigenvalues
Date   Fri, 15 Apr 2005 22:03:18 +0900

Dear Statalisters,

I'm working with transition matrices (namely, the matrices 
whose rows all sum up to be unity) of 47 x 47 size for 20 years
using Stata 8. 
And I have two questions as to this topic.

(1)
As is well known, their maximal eigenvalues are guaranteed 
to be unity.
And one way to measure the speed of convergence to the long-run 
equilibrium state of the given transition matrix is to use 
the absolute value of the 2nd (largest) eigenvalue.
So, it'd be convenient to sort eigenvalues in descending (or 
ascending) order for each matrix and (if possible) extract 
the 2nd eigenvalues alone to combine/graph them and check 
their time-series patten.
# Currently I copy and paste each value from a log file.
Is there any handy way to do so in Stata?

(2)
Regarding the above point, though the Programming Reference 
Manual says that the extracted eigenvalues are to be sorted 
in ascending order, it's not necessarily the case; when the 
codes below (the ones suggested in the Manual) are issued,

forvalues i = 1/47 { 
	di sqrt(re[1, `i']^2 + im[1,`i']^2)
}

following outputs are shown (sorry for lengthy texts): 

.40468505
.50863593
.57787482
.66516332
.70771485
.84072082
.87075128
.89745642
.90354965
.90619857
.91547048
.9183964
.92652875
.92758641
.93138665
.9430583
.94063044
.94064868
.95091566
.94943396
.95769209
.96646326
.97100423
.97126325
.99999999
.99876833
.97456569
.97597157
.97597157
.97759933
.97843964
.97892578
.98151891
.98151891
.99748731
.99651247
.9962227
.99432542
.99370197
.99310467
.98430497
.98498111
.98664358
.9900003
.98993781
.98887064
.98861453

You can check that these values are not necessarily 
monotonically increasing.
Is this just a coincidence or an environment-specific 
phenomenon (or something else) ?

Any suggestions welcome.
Thanks in advance.

K.I.
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