On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property (Q1947189)

The paper deals with the Gaussian elimination of completely pivoted Hadamard matrices. The stability of Gaussian elimination depends on the growth factor which involves all elements of the matrix \(A\) that occur during the elimination. The authors recall Hadamard equivalent matrices and Cryer's growth conjecture. At the end of the introduction, they say that ``research on the values of minors of Hadamard matrices is still ongoing.'' The authors prove that the leading principal minors of a completely pivoted Hadamard matrix form an increasing sequence. They give the bounds of the growth factor for a completely pivoted Hadamard matrix of order 6 and 7 and give a new proof that the growth of a Hadamard matrix of order 12 equals 12. The authors introduce so-called ``good pivot patterns'' and say that ``so far Hadamard matrices are the only matrices known that attain this property.'' They introduce also an infinite family of Hadamard matrices with goods pivots.