From: Globe Trotter <itsme_410_at_yahoo.com>

Date: Tue 03 May 2005 - 06:50:23 EST

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R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Tue May 03 06:55:43 2005

Date: Tue 03 May 2005 - 06:50:23 EST

The example that I submitted earlier in the day. Would you like me to send
again?

Thanks!

- "Huntsinger, Reid" <reid_huntsinger@merck.com> wrote:

> For which X?

*>
**> Reid Huntsinger
**>
**> -----Original Message-----
**> From: Globe Trotter [mailto:itsme_410@yahoo.com]
**> Sent: Monday, May 02, 2005 2:34 PM
**> To: Huntsinger, Reid; Rolf Turner
**> Cc: r-help@stat.math.ethz.ch
**> Subject: RE: [R] eigenvalues of a circulant matrix
**>
**>
**> By the way, I just noticed that eigen(X) returns eigenvectors, at least two
**> of
**> which are NaN's.
**>
**> Best wishes!
**>
**> --- "Huntsinger, Reid" <reid_huntsinger@merck.com> wrote:
**>
**> > When the matrix is symmetric and omega is not real, omega and its
**> conjugate
**> > (= inverse) give the same eigenvalue, so you have a 2-dimensional
**> > eigenspace. R chooses a real basis of this, which is perfectly fine since
**> > it's not looking for circulant structure.
**> >
**> > For example,
**> >
**> > > m
**> > [,1] [,2] [,3] [,4] [,5]
**> > [1,] 1 2 3 3 2
**> > [2,] 2 1 2 3 3
**> > [3,] 3 2 1 2 3
**> > [4,] 3 3 2 1 2
**> > [5,] 2 3 3 2 1
**> >
**> > > eigen(m)
**> > $values
**> > [1] 11.000000 -0.381966 -0.381966 -2.618034 -2.618034
**> >
**> > $vectors
**> > [,1] [,2] [,3] [,4] [,5]
**> > [1,] 0.4472136 0.000000 -0.6324555 0.6324555 0.000000
**> > [2,] 0.4472136 0.371748 0.5116673 0.1954395 0.601501
**> > [3,] 0.4472136 -0.601501 -0.1954395 -0.5116673 0.371748
**> > [4,] 0.4472136 0.601501 -0.1954395 -0.5116673 -0.371748
**> > [5,] 0.4472136 -0.371748 0.5116673 0.1954395 -0.601501
**> >
**> > and you can match these columns up with the "canonical" eigenvectors
**> > exp(2*pi*1i*(0:4)*j/5) for j = 0,1,2,3,4. E.g.,
**> >
**> > > Im(exp(2*pi*1i*(0:4)*3/5))
**> > [1] 0.0000000 -0.5877853 0.9510565 -0.9510565 0.5877853
**> >
**> > which can be seen to be a scalar multiple of column 2.
**> >
**> > Reid Huntsinger
**> >
**> > Reid Huntsinger
**> >
**> > -----Original Message-----
**> > From: r-help-bounces@stat.math.ethz.ch
**> > [mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Huntsinger, Reid
**> > Sent: Monday, May 02, 2005 10:43 AM
**> > To: 'Globe Trotter'; Rolf Turner
**> > Cc: r-help@stat.math.ethz.ch
**> > Subject: RE: [R] eigenvalues of a circulant matrix
**> >
**> >
**> > It's hard to argue against the fact that a real symmetric matrix has real
**> > eigenvalues. The eigenvalues of the circulant matrix with first row v are
**> > *polynomials* (not the roots of 1 themselves, unless as Rolf suggested you
**> > start with a vector with all zeros except one 1) in the roots of 1, with
**> > coefficients equal to the entries in v. This is the finite Fourier
**> transform
**> > of v, by the way, and takes real values when the coefficients are real and
**> > symmetric, ie when the matrix is symmetric.
**> >
**> > Reid Huntsinger
**> >
**> > -----Original Message-----
**> > From: r-help-bounces@stat.math.ethz.ch
**> > [mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Globe Trotter
**> > Sent: Monday, May 02, 2005 10:23 AM
**> > To: Rolf Turner
**> > Cc: r-help@stat.math.ethz.ch
**> > Subject: Re: [R] eigenvalues of a circulant matrix
**> >
**> >
**> >
**> > --- Rolf Turner <rolf@math.unb.ca> wrote:
**> > > I just Googled around a bit and found definitions of Toeplitz and
**> > > circulant matrices as follows:
**> > >
**> > > A Toeplitz matrix is any n x n matrix with values constant along each
**> > > (top-left to lower-right) diagonal. matrix has the form
**> > >
**> > > a_0 a_1 . . . . ... a_{n-1}
**> > > a_{-1} a_0 a_1 ... a_{n-2}
**> > > a_{-2} a_{-1} a_0 a_1 ... .
**> > > . . . . . .
**> > > . . . . . .
**> > > . . . . . .
**> > > a_{-(n-1)} a_{-(n-2)} ... a_1 a_0
**> > >
**> > > (A Toeplitz matrix ***may*** be symmetric.)
**> >
**> > Agreed. As may a circulant matrix if a_i = a_{p-i+2}
**> >
**> > >
**> > > A circulant matrix is an n x n matrix whose rows are composed of
**> > > cyclically shifted versions of a length-n vector. For example, the
**> > > circulant matrix on the vector (1, 2, 3, 4) is
**> > >
**> > > 4 1 2 3
**> > > 3 4 1 2
**> > > 2 3 4 1
**> > > 1 2 3 4
**> > >
**> > > So circulant matrices are a special case of Toeplitz matrices.
**> > > However a circulant matrix cannot be symmetric.
**> > >
**> > > The eigenvalues of the forgoing circulant matrix are 10, 2 + 2i,
**> > > 2 - 2i, and 2 --- certainly not roots of unity.
**> >
**> > The eigenvalues are 4+1*omega+2*omega^2+3*omega^3.
**> > omega=cos(2*pi*k/4)+isin(2*pi*k/4) as k ranges over 1, 2, 3, 4, so the
**> above
**> > holds.
**> >
**> > Bellman may have
**> > > been talking about the particular (important) case of a circulant
**> > > matrix where the vector from which it is constructed is a canonical
**> > > basis vector e_i with a 1 in the i-th slot and zeroes elsewhere.
**> >
**> > No, that is not true: his result can be verified for any circulant matrix,
**> > directly.
**> >
**> > > Such a matrix is in fact a unitary matrix (operator), whence its
**> > > spectrum is contained in the unit circle; its eigenvalues are indeed
**> > > n-th roots of unity.
**> > >
**> > > Such matrices are related to the unilateral shift operator on
**> > > Hilbert space (which is the ``primordial'' Toeplitz operator).
**> > > It arises as multiplication by z on H^2 --- the ``analytic''
**> > > elements of L^2 of the unit circle.
**> > >
**> > > On (infinite dimensional) Hilbert space the unilateral shift
**> > > looks like
**> > >
**> > > 0 0 0 0 0 ...
**> > > 1 0 0 0 0 ...
**> > > 0 1 0 0 0 ...
**> > > 0 0 1 0 0 ...
**> > > . . . . . ...
**> > > . . . . . ...
**> > >
**> > > which maps e_0 to e_1, e_1 to e_2, e_2 to e_3, ... on and on
**> > > forever. On (say) 4 dimensional space we can have a unilateral
**> > > shift operator/matrix
**> > >
**> > > 0 0 0 0
**> > > 1 0 0 0
**> > > 0 1 0 0
**> > > 0 0 1 0
**> > >
**> > > but its range is a 3 dimensional subspace (e_4 gets ``killed'').
**> > >
**> > > The ``corresponding'' circulant matrix is
**> > >
**> > > 0 0 0 1
**> > > 1 0 0 0
**> > > 0 1 0 0
**> > > 0 0 1 0
**> > >
**> > > which is an onto mapping --- e_4 gets sent back to e_1.
**> > >
**> > > I hope this clears up some of the confusion.
**> > >
**> > > cheers,
**> > >
**> > > Rolf Turner
**> > > rolf@math.unb.ca
**> >
**> > Many thanks and best wishes!
**> >
**> > ______________________________________________
**> > R-help@stat.math.ethz.ch mailing list
**> > https://stat.ethz.ch/mailman/listinfo/r-help
**> > PLEASE do read the posting guide!
**> > http://www.R-project.org/posting-guide.html
**> >
**> > ______________________________________________
**> > R-help@stat.math.ethz.ch mailing list
**> > https://stat.ethz.ch/mailman/listinfo/r-help
**> > PLEASE do read the posting guide!
**> > http://www.R-project.org/posting-guide.html
**> >
**> >
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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Tue May 03 06:55:43 2005

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