Matrix circulant

By relabeling each of the cyclic blocks, Ci, in such a way that the transition diagram proceeds systematically from state 1 to Cj, Cj may be rewritten as a circulant matrix with first row = (0,0,..., 1) ... 

CDC 6000, 210 COPAS, 217 Caruso, 134 Cepstral distance, 30 Chaos measure, 61 Characteristic impedance, 433 Chipmunk effect, 322 Cholesky decomposition, 148 Chorusing, 298, 303 Circulant matrix, 126 Clarinet synthesis, 455 Clarity index, 98 Clicks (See Restoration)... 

Each sequence is then expressed as an infinite circulant matrix whose columns (or the reverses of the rows) are all copies of the sequence. This analogue is most relevant when one or both of the sequences is potentially infinite in length. 

We can reexpress this matrix as the product of two matrices. The first is a circulant matrix with every row29 and every column a copy of the mask divided by 2. The second is a diagonal matrix with the value 2 on the diagonal of slope -2. We call the latter the sampling matrix and the former the smoothing matrix. 

The smoothing matrix can now be viewed as a filter which will have different effects on the two components. We attack the analysis of its effects by doing a further factorisation, exploiting the fact that the z transform can be regarded as a representation of the smoothing matrix as a polynomial in the unit subdiagonal. Factorisation of a circulant matrix is exactly equivalent to factorisation of the generating function as a polynomial. 

This can be traced back to the result that the two-slanted sampling matrix S encountered in the chapter on artifacts has the same product when multiplied on the right by the circulant matrix corresponding to (1 + z) as when multiplied on the left by that corresponding to (1 + z2). 

The unit row eigenvector is a stencil which gives a point on the limit curve in terms of the original control points. The product of a circulant matrix, E, all of whose rows are equal to that eigenvector, with the control polygon, P, gives a sequence of points, Q, on the limit curve. 

A circulant matrix is thus completely defined once the elements in its first row have been specified furthermore, the eigenvalues and eigenvectors of a circulant may be obtained analytically in closed form, as we shall now show. 

The circulant matrix n is a special case of a permutation matrix. In the above example, the P matrix is the shift matrix n (0, 1, 0). 

G8. Gohji, K., Fujimoto, N., Fujii, A., Komiyama, T., Okawa, J., and Nakajima, M., Prognostic significance of circulating matrix metalloproteinase-2 to tissue inhibitor of metalloproteinases-2 ratio in recurrence of urothelial cancer after complete resection. Cancer Res. 56, 3196-3198 (1996). 

K2. Kasahara, A., Hayashi, N., Mochizuki, K., et al. Circulating matrix metalloproteinase-... 

N8. Noji, Y., Kajinami, K., Kawashiri, M. A., et at. Circulating matrix metalloproteinases and their inhibitors in premature coronary atherosclerosis. Clin. Chem. Lab. Med. 39, 380-384 (2001). 

Circulant matrix A ToepHtz matrix in which every row beginning from the second is obtained from the proceeding row through a cycHc shift. 

Koos R, Krueger T, Westenfeld R, et al. Relation of circulating Matrix Gla-Protein and anticoagulation status in patients with aortic valve calcification, Thromb Haemost 2009 101 706-713. 

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