On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property

Authors:
Christos Kravvaritis;Marilena Mitrouli
Affiliations:
Department of Mathematics, University of Athens, Athens, Greece 15784;Department of Mathematics, University of Athens, Athens, Greece 15784
Venue:
Numerical Algorithms
Year:
2013

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Abstract

Further progress is achieved for the growth conjecture for Hadamard matrices. It is proved that the leading principal minors of a CP Hadamard matrix form an increasing sequence. Bounds for the sixth and seventh pivot of any CP Hadamard matrix are given. A new proof demonstrating that the growth of a Hadamard matrix of order 12 is 12, is presented. Moreover, a new notion of good pivots is introduced and its importance for the study of the growth problem for CP Hadamard matrices is examined. We establish that CP Hadamard matrices with good pivots satisfy Cryer's growth conjecture with equality, namely their growth factor is equal to their order. A construction of an infinite class of Hadamard matrices is proposed.