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.) 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. 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. 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] ______________________________________________ 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