Matrices Of Odd Order Research Articles

A Fast Recursive Algorithm for Multiplying Matrices of Order n = 3q (q > 1)

A new fast recursive algorithm is proposed for multiplying matrices of order n = 3q (q > 1). This algorithm is based on the hybrid algorithm for multiplying matrices of odd order n = 3μ (μ = 2q – 1, q > 1), which is used as a basic algorithm for μ = 3q (q > 0). As compared with the well-known block-recursive Laderman’s algorithm, the new algorithm minimizes by 10.4% the multiplicative complexity equal toWm = 0 896n2.854 multiplication operations at recursion level d = log3 n – 3 and reduces the computation vector by three recursion steps. The multiplicative complexity of the basic and recursive algorithms are estimated.

A proof of non-existence of bordered jacket matrices of odd order over some fields

Lee and colleagues have introduced a generalised reverse jacket transform (GRJT) as a multi-phase or multilevel generalisation of the WHT and the even-length DFT. The matrix representing a primary GRJT is permutation-equivalent to the DFT matrix of even order, and in addition possesses the property to have a border consisting entirely of ±1's. To the authors' knowledge, the problem of existence of such transform matrices in the case of odd order, has attracted minor attention in the existing literature. In this Letter, a contribution is made regarding this problem in respect to an extended class of matrices.

On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order—I

On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order—I

On circulant best matrices and their applications

Call four type 1(1,−1) matricesX 1 X 2 X 3 X 4, on the same group of order m (odd) with the properties (i) (Xt −I)T=−(Xt −I)i=l, 2, 3, (ii) and the diagonal elements are positive, (iii) Xi Xj =Xi Xj and (iv) , best matrices. We use a computer to give, for the first time, all inequivalent best matrices of odd order m⩽31. Inequivalent best matrices of order m m odd, can be used to find inequivalent skew-Hadamard matrices of order 4m. We use best matrices of order (1/4) (s 2+3) to construct new orthogonal designs, including new OD(2s 2+6;1,1,2,2,s 2 s 2).