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By Philip J. Davis
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Additional resources for Circulant Matrices
Let F 1 designate the Fourier matrices of order n 2 2 whose rows have been permuted according to the bit reversing permutation (see Problem 6 , p. 30). 15) where F^ = (I2 ® F'2)D 4 (F2 0 I2) = diag(l, 1, 1, i). This may be easily checked out. As is known, A ® B = P(B 0 A)P* for some permu tation matrix P that depends merely on the dimensions of A and B. 16) (one has, in fact, F 4 = (I2 Similarly, 0 = S^): F£)D4 S 4 (I2 0 F^)S4. 19) D = diag (1, w, w 2 , w3) , w = exp — . 20) F '6 = ( I 4 6 F ' ) D 1 6 S1 6 ( I 4 ® n ) S 1 6 .
K J. 4 f NESTED POLYGONS We pass now from triangles to polygons. , Let z^, z^, be ordered vertices of a polygon P (assumed to be located in the complex plane). We make no restric tions on the complex numbers z s o that P may be con vex or nonconvex, simply covered or not; furthermore, the points z^. " All geometric constructions described below are to be interpreted appropriately with this in mind. We shall also call such a figure a p-gon. We shall assume, however, that the center of gravity of P, 1/pCz^ + ...
CO Corollary. 1 \ 1 \ 0 ... 7) F = 7l . (It may be shown that all the qth roots of I are of the form M 1DM where D = diagi]^, p2 , •••* U ) / y? ) Corollary. The eigenvalues of F are ±1, ±i, with appropriate multiplicities. Carlitz has obtained the characteristic polynom ials f (A) of F* (= F*). They are as follows. n = 0 (mod 4), f(X) = (X - 1)2 (X - i)(X + 1) (x4 - i)(n/4> - \ Introductory Matrix Material 34 n = 1 (mod 4), f (A ) = (A - 1) (A4 - i) ( V 4 ) ( n - D n = 2 (mod 4), f (A ) = (A* 2 - 1) (A4 - i) d / 4 ) (n-2) ^ n = 3 (mod 4), f (A) = (A - i) (A2 - 1) ' (A4 - D d / i X n - S ) .