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power of a matrix

2 messages · Rau, Roland, Dimitris Rizopoulos

Original
Rau, Roland
# 
Dear all,

I have a population with three age-classes, at time t=0 the population
is:
n.zero <- c(1,0,0)
I have a transition matrix A which denotes "fertility" and "survival":

A <- matrix(c(0,1,5, 0.3,0,0, 0,0.5,0), ncol=3, byrow=TRUE)

To obtain the population at t=1, I calculate:
A %*% n.zero

To obtain the population t=2, I calculate:
A %*% (A %*% n.zero)

... and so on ...

I thought now to obtain the population at time x, I should simply do:
A^x %*% n.zero

But this, of course, does not work since the following two statements
are not equivalent for matrices:
A %*% A
A * A

Is there something like a "powermatrix"-function?

Thanks,
Roland

P.S. The example is taken from:
Example 3.1 ("A linear, time-invariant model") from Keyfitz/Caswell:
"Applied Mathematical Demography", 3rd Edition, 2005, p. 52f


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Dimitris Rizopoulos
# 
look at function ?mtx.exp() in the Malmig package, e.g.,

A <- matrix(c(0,1,5, 0.3,0,0, 0,0.5,0), ncol=3, byrow=TRUE)
n.zero <- c(1,0,0)
##################
library(Malmig)

all.equal(A %*% (A %*% n.zero), mtx.exp(A, 2) %*% n.zero)
all.equal(A %*% (A %*% (A %*% n.zero)), mtx.exp(A, 3) %*% n.zero)

mtx.exp(A, 15) %*% n.zero


I hope it helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/16/336899
Fax: +32/16/337015
Web: http://www.med.kuleuven.be/biostat/
     http://www.student.kuleuven.be/~m0390867/dimitris.htm


----- Original Message ----- 
From: "Rau, Roland" <Rau at demogr.mpg.de>
To: "R-Help" <r-help at stat.math.ethz.ch>
Sent: Wednesday, August 17, 2005 12:15 PM
Subject: [R] power of a matrix