We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for |q0| = . . . = |qN−1| = 1 the quantity Φ = ∑ i+k=j+l qiqk qjql satisfies Φ ≥ N, with equality if and only if q = (qi) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems,...

The necessary condition for eigenvalue values of a circulant matrix is studied It is then proved that the necessary condition also su ces the existence of a circulant matrix with the prescribed eigenvalue values Introduction An n n matrix C of the form C c c cn cn c c cn cn cn c cn c c cn c is called a circulant matrix As each row of a circulant matrix is just the previous row cycled forward on...

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J × L exponent matrix E, the lower bound Q(E) is obtained explicitly such that (J, L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ≥ Q(E). The results show that the exponent matrices constructed with ou...

In this paper, we use the algebra methods, the properties of the r-circulant matrix and the geometric circulant matrix to study the upper and lower bound estimate problems for the spectral norms of a geometric circulant matrix involving the generalized k-Horadam numbers, and we obtain some sharp estimations for them. We can also give a new estimation for the norms of a r-circulant matrix involv...

In this paper, the notions of pseudofield of values and joint pseudofield of values of matrix polynomials are introduced and some of their algebraic and geometrical properties are studied.  Moreover, the relationship between the pseudofield of values of a matrix polynomial and the pseudofield of values of its companion linearization is stated, and then some properties of the augmented field of ...

The permanent of an n x n matrix A = (a;j) is the matrix function ( 1) per A = ∑ al1r(1)••• a .. ",( .. )".C~" where the summation is over all permutations in the symmetric group, S ... An n x n matrix A is a circulant if there are scalars ab ... ,a,. such that (2) A= ∑ a;pi-l where P is the n x n permutation matrix corresponding to the cycle (12• .. n) in s". In general the computation of the ...

Preconditioned conjugate gradient method is used to solve n-by-n Hermitian Toeplitz systems A n x = b. The preconditioner S n is the Strang's circulant preconditioner which is deened to be the circulant matrix that copies the central diagonals of A n. The convergence rate of the method depends on the spectrum of S ?1 n A n. Using Jackson's theorem in approximation theory, we prove that if A n h...

A circulant matrix of order n is the matrix of convolution by a fixed element of the group algebra of the cyclic group Zn. Replacing Zn by an arbitrary finite group G gives the class of matrices that we call G-circulant. We determine the eigenvalues of such matrices with the tools of representation theory and the non-abelian Fourier transform. Definition 1 An n by n matrix C is circulant if the...

The main aim of this paper is to introduce three classes $H^0_{p,q}$, $H^1_{p,q}$ and $TH^*_p$ of $p$-harmonic mappings and discuss the properties of mappings in these classes. First, we discuss the starlikeness and convexity of mappings in $H^0_{p,q}$ and $H^1_{p,q}$. Then establish the covering theorem for mappings in $H^1_{p,q}$. Finally, we determine the extreme points of the class $TH^*_{p}$.

For simplicity, we adopt the following convention: i, j, k, n, l denote elements of N, K denotes a field, a, b, c denote elements of K, p, q denote finite sequences of elements of K, and M1, M2, M3 denote square matrices over K of dimension n. Next we state two propositions: (1) 1K · p = p. (2) (−1K) · p = −p. Let K be a set, let M be a matrix over K, and let p be a finite sequence. We say that...