Re: roots of determinant of toeplitz-like matrix

From: David Belohrad (david.belohrad_at_cern.ch)
Date: 09/02/04 

Date: Thu, 02 Sep 2004 10:41:15 +0200

Hi Robert,
thanks for response, I need to think a bit of this. But generaly,
perhaps I miss something, but I would not say that I'm looking for
eigenvector of this matrix. while you're solving the
equation det (M-lambda I) = 0 -> lambda,
I'm looking for det(M) = 0 -> omega (s=I*omega, where s is laplace
operator and I is complex number, so the determinant is in the form
of a*s^n+b*s^(n-1)....+z).
In the result we receive the polynoms of the same order, but
the meaning is different (am i wrong?) 

---
generaly I'm not interested in full solution of the system. I'm looking
for the lowest real root, which gives the basic resonant frequency.
---
....not solvable by radicals
does it mean that you can use numeric solutions only?
thanks a lot
david
Robert Israel wrote:
> In article <41364f4d$0$141$fb624d75@newsspool.solnet.ch>,
> dejfson  <dejfson@solnet.ch> wrote:
> 
> 
>>I have a matrix M of N-th order (N=10..15), which looks like a toeplitz 
>>matrix with the structure (A, -1, 0 .... 0) (in diagonal is A, just one 
>>below and above diagonal is -1, others are zeroed). The element M[1,1] 
>>and M[N,N] are different from A (can be whatever complex).
> 
> 
> So for N=4, if I understand you, the matrix looks like
> 
> [  x -1  0  0 ] 
> [ -1  A -1  0 ]
> [  0 -1  A -1 ]
> [  0  0 -1  y ]
> 
> where A, x and y are arbitrary.
> 
> 
> 
>>---
>>In principle this matrix describes the behaviour of electronic circuit. 
>>What I need is to find out the resonant frequencies of the circuit. 
>>These are given by the roots of the determinant of the matrix. The 
> 
> 
> I think you mean the eigenvalues.  These are the roots of the 
> characteristic polynomial P(lambda), which is det(M - lambda I) 
> where I is the identity matrix.  Unfortunately, if you're looking for
> closed-form solutions, I think you're out of luck: e.g. for n=5,
> x=1,y=2,A=0, the characteristic polynomial is 
> lambda^5 - 3 lambda^4 - 2 lambda^3 + 9 lambda^2 - lambda - 3,
> which has Galois group S_5 and thus is not solvable by radicals.
> 
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            
> Vancouver, BC, Canada V6T 1Z2
> 
> 
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