By the way, I just noticed that eigen(X) returns eigenvectors, at least two of which are NaN's.
Best wishes! --- "Huntsinger, Reid" <[EMAIL PROTECTED]> 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: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] 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: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] 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 <[EMAIL PROTECTED]> 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 > > [EMAIL PROTECTED] > > 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 > > ---------------------------------------------------------------------------- > -- > Notice: This e-mail message, together with any attachments, contains > information of Merck & Co., Inc. 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