On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

D-optimal designs embedded in Hadamard matrices and their effect on the pivot patterns

In this paper we develop a new approach for detecting if specific D-optimal designs exist embedded in Sylvester-Hadamard matrices. Specifically, we investigate the existence of the D-optimal designs of orders 5, 6, 7 and 8. The problem is motivated to explaining why specific values appear as pivot elements when Gaussian elimination with complete pivoting is applied to Hadamard matrices. Using this method and a complete search algorithm we explain, for the first time, the appearance of concrete pivot values for equivalence classes of Hadamard matrices of orders n = 12, 16 and 20.     

On the computation of the Smith normal form of compound matrices

Compound matrices are encountered in many fields such as Matrix Theory, Systems Theory, Control Theory, etc. In the present paper we develop an efficient algorithm computing the Smith normal form of compound matrices. This algorithm is based on a new theorem establishing an equivalence relation between the Smith normal form of the compounds of a given matrix and the compounds of the Smith normal form of the given matrix. This revised version was published online in June 2006 with corrections to the Cover Date.     

An Infinite Family of Hadamard Matrices with Fourth Last Pivot n /2

We show that the equivalence class of Sylvester Hadamard matrices give an infinite family of Hadamard matrices in which the fourth last pivot is n /2. Analytical examples of completely pivoted Hadamard matrices of order n having as fourth last pivot n /2 are given for n = 16 and 32. In each case this distinguished case with the fourth pivot n /2 arose in the equivalence class of the Sylvester Hadamard matrices.     

On rank and null space computation of the generalized Sylvester matrix

In this paper, we present new approaches computing the rank and the null space of the (m n + p)×(n + p) generalized Sylvester matrix of (m + 1) polynomials of maximal degrees n,p. We introduce an algorithm which handles directly a modification of the generalized Sylvester matrix, computing efficiently its rank and null space and replacing n by log ₂ n in the required complexity of the classical methods. We propose also a modification of the Gauss-Jordan factorization method applied to the appropriately modified Sylvester matrix of two polynomials for computing simultaneously its rank and null space. The methods can work numerically and symbolically as well and are compared in respect of their error analysis, complexity and efficiency. Applications where the computation of the null space of the generalized Sylvester matrix is required, are also given.     

Compound matrices: properties,numerical issues and analytical computations

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.      

A compound matrix algorithm for the computation of the Smith form of a polynomial matrix

In the present paper is presented a numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations. Using the notion of compound matrices, the Smith canonical form of a polynomial matrixM(s)^nxn[s] is calculated directly from its definition, requiring only the construction of all thep-compound matricesC _p (M(s)) ofM(s), 1<pn. This technique produces a stable and accurate numerical algorithm working satisfactorily for any polynomial matrix of any degree.     

On the computation of the rank of block bidiagonal Toeplitz matrices

In the present paper we study the computation of the rank of a block bidiagonal Toeplitz (BBT) sequence of matrices. We propose matrix-based, numerical and symbolical, updating and direct methods, computing the rank of BBT matrices and comparing them with classical procedures. The methods deploy the special form of the BBT sequence, significantly reducing the required flops and leading to fast and efficient algorithms. The numerical implementation of the algorithms computes the numerical rank in contrast with the symbolical implementation, which guarantees the computation of the exact rank of the matrix. The combination of numerical and symbolical operations suggests a new approach in software mathematical computations denoted as hybrid computations.     

An algorithm to find formulae and values of minors for Hadamard matrices

We give an algorithm to obtain formulae and values for minors of Hadamard matrices. One step in our algorithm allows the (n−j)×(n−j) minors of a Hadamard matrix to be given in terms of the minors of a 2^j−1×2^j−1 matrix. In particular we illustrate our algorithm by finding explicitly all the (n−4)×(n−4) minors of a Hadamard matrix.