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

Koukouvinos, Christos; Lappas, E.; Mitrouli, Marilena; Seberry, Jennifer

doi:10.1016/s0024-3795(01)00249-x

Cited by 19 publications

(13 citation statements)

References 2 publications

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“…Proof. As already noted in [12], these formulae from [6], [7] follow from Theorem 2.3. They are even more clear from the above formulae of S (1) and S (2) , using the fact that the non-zero eigenvalues of…”

Section: Complements Of Small Sign Patternssupporting

confidence: 59%

Submatrices of Hadamard matrices: complementation results

Banica¹,

Nechita

Schlenker³

2014

ELA

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Two submatrices A, D of a Hadamard matrix H are called complementary if, up to a permutation of rows and columns, H = [ A C B D ]. We find here an explicit formula for the polar decomposition of D. As an application, we show that under suitable smallness assumptions on the size of A, the complementary matrix D is an almost Hadamard sign pattern, i.e. its rescaled polar part is an almost Hadamard matrix. 2000 Mathematics Subject Classification. 15B34.

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Section: Complements Of Small Sign Patternssupporting

confidence: 59%

Submatrices of Hadamard matrices: complementation results

Banica¹,

Nechita

Schlenker³

2014

ELA

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“…For more information on minors of Hadamard matrices we refer to the articles [10], [11], see also [4], where square submatrices of Hadamard matrices are considered.…”

Section: Conjecture 31 Every Partial Hadamard Matrix In H ∈ M 4×5 (mentioning

confidence: 99%

The quantum algebra of partial Hadamard matrices

Banica

Skalski

2015

Linear Algebra and its Applications

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Abstract. A partial Hadamard matrix is a matrix H ∈ M M×N (T) whose rows are pairwise orthogonal. We associate to each such H a certain quantum semigroup G of quantum partial permutations of {1, . . . , M } and study the correspondence H → G. We discuss as well the relation between the completion problems for a given partial Hadamard matrix and completion problems for the associated submagic matrix P ∈ M M (M N (C)), in both cases introducing certain criteria for the existence of the suitable completions.

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“…The first known effort for calculating minors of Hadamard matrices was accomplished in [20] for the n −1, n −2 and n −3 minors in a totally different manner than the one presented here. In [21], a method for evaluating all possible (n − j)×(n − j) minors of Hadamard matrices was developed theoretically, which could be generalized as an algorithm. In the present paper this technique was appropriately modified in order to work more effectively and to deal better with the particular problem.…”

Section: An Algorithm Computing (N − J)×(n − J) Minors Of Hadamard Mamentioning

confidence: 99%

The growth factor of a Hadamard matrix of order 16 is 16

Kravvaritis

Mitrouli

2009

Numerical Linear Algebra App

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In 1968 Cryer conjectured that the growth factor of an n ×n Hadamard matrix is n. In 1988 Day and Peterson proved this only for the Hadamard-Sylvester class. In 1995 Edelman and Mascarenhas proved that the growth factor of a Hadamard matrix of order 12 is 12. In the present paper we demonstrate the pivot structures of a Hadamard matrix of order 16 and prove for the first time that its growth factor is 16. The study is divided in two parts: we calculate pivots from the beginning and pivots from the end of the pivot pattern. For the first part we develop counting techniques based on symbolic manipulation for specifying the existence or non-existence of specific submatrices inside the first rows of a Hadamard matrix, and so we can calculate values of principal minors. For the second part we exploit sophisticated numerical techniques that facilitate the computations of all possible (n − j)×(n − j) minors of Hadamard matrices for various values of j. The pivot patterns are obtained by utilizing appropriately the fact that the pivots appearing after the application of Gaussian elimination on a completely pivoted matrix are given as quotients of principal minors of the matrix.

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

Cited by 19 publications

References 2 publications

Submatrices of Hadamard matrices: complementation results

Submatrices of Hadamard matrices: complementation results

The quantum algebra of partial Hadamard matrices

The growth factor of a Hadamard matrix of order 16 is 16

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