How about the following:
library(polynom)
help(package="polynom")
A <- diag(c(1:2, 2))
eigVals <- eigen(A)$values
multEig <- table(eigVals)
k <- length(multEig)
ratPoly <- minPoly <- 1
for(i in 1:k){
poly.i <- polynomial(c(-as.numeric(names(multEig)[i]), 1))
minPoly <- (minPoly*poly.i)
if(multEig[i]>1)
ratPoly <- (ratPoly*poly.i^(multEig[i]-1))
}
> minPoly
2 - 3*x + x^2
> ratPoly
-2 + x
>
hope this helps. spencer graves
###############################
Spencer,
One could do this by a brute force approach. Suppose A is an nxn matrix, and
the distinct eigenvalues are: lambda_1, ..., lambda_k, with multiplicities
m_1, ..., m_k, such that they sum to n. Then the characteristic polynomial
is:
C(lambda) = \prod_i (lambda - lambda_i)^{m_i}
The minimal polynomial is given by:
M(lambda) = \prod_i (lambda - lambda_i)^{p_i},
where 1 \leq p_i \leq m_i.
So, one could run through all possible p_i, starting with the smallest
degree polynomial (within constraint), and stop when we find one that
exactly divides C(lambda).
Is there a cleverer way to do this?
Ravi.
#################################################
Have you looked at library(polynom)? Will that with
unique(eigen(A)$values) allow you to compute what you want?
hope this helps.
spencer graves
Ravi Varadhan wrote:
Hi,
I would like to know whether there exist algorithms to compute the
coefficients or, at least, the degree of the minimal polynomial of a square
matrix A (over the field of complex numbers)? I don't know whether this
would require symbolic computation. If not, has any of the algorithms been
implemented in R?
Thanks very much,
Ravi.
P.S. Just for the sake of completeness, a minimal polynomial is a monic
polynomial (whose leading coefficient is unity) of least degree, which
divides all the annihilating polynomial of A. In particular, the minimal
polynomial divides the characteristic polynomial. Knowing the degree of the
minimal polynomial is useful in characterizing the convergence properties of
a certain class of numerical schemes for iteratively solving linear (and
nonlinear) system of equations.
--------------------------------------------------------------------------
Ravi Varadhan, Ph.D.
Assistant Professor, The Center on Aging and Health
Division of Geriatric Medicine and Gerontology
Johns Hopkins Univerisity
Ph: (410) 502-2619
Fax: (410) 614-9625
Email: <mailto:[EMAIL PROTECTED]> [EMAIL PROTECTED]
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