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X.—Note on Pfaffians with Polynomial Elements
Published online by Cambridge University Press: 15 September 2014
Extract
So far back as March 1855 Brioschi in effect formulated the theorem that any even-ordered determinant is expressible as a Pfaffian; for example in the case of the fourth order he gave the equality
where (h, k) stands for
this expression being obtained by multiplying the hth column
of the given determinant by the kth column
of an equivalent determinant.
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- Copyright © Royal Society of Edinburgh 1920
References
page 86 note * This is at variance with a statement of Pascal's when dealing with Brioschi's theorem above referred to. In his paper on “Un teorema sui determinanti di ordine pari,” he says, “Gli altri termini dello sviluppo del prodotto (7) sono similmente termini dello sviluppo di determinanti come D dove però alcune coppie di linee sono ripetute e che quindi sono zero” (see Rendic … Accad … Napoli, xxv, 1919). The case where m is 2 in § 3 provides the simplest test: for then the number of terms on the left is twelve, and the number on the right is 6, and the 6 on the left whose aggregate must vanish are
which, instead of forming a vanishing determinant, form two vanishing Pfaffians.
page 87 note * Trans. S. African Phil. Soc., xv, pp. 35–41, §§ 6, 7.
page 88 note * In view of the footnote above (§ 7), it is fitting that I should here point out an incorrect footnote of my own bearing on a determinant and a sum of Pfaffians. See Trans. Roy. Soc. Edin., xlv, 1906-7, p. 313.