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XXII.—Vanishing Aggregates of Secondary Minors of a Persymmetric Determinant

Published online by Cambridge University Press:  06 July 2012

Thomas Muir
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Extract

(1) The persymmetric determinant of the nth order

being such that in every case the element in the place r, s is the same as the element in the place r − 1, s + 1, and therefore having only 2n − 1 independent elements, viz., the elements a1, a2, …., a2n−1 forming the first row and last column, is conveniently denoted by

As it is a special form of axisymmetric determinant, any known relation between minors of the latter must of course hold in regard to the corresponding minors of the former.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 40 , Issue 3 , 1905 , pp. 511 - 533
Copyright
Copyright © Royal Society of Edinburgh 1905

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References

page 511 note * Muir, Th., “The automorphic linear transformation of a quadric,” Trans. Roy. Soc. Edin., xxxix. [pp. 209–230] p. 226Google Scholar.

page 511 note * Cazzaniga, Tito, “Relazioni fra i minori di un determinante di Hankel,” Rendiconti del R. Ist. Lomb. di sc., e lett., serie ii. vol. xxxi. (1898) pp. 610–614Google Scholar.

page 525 note * As I have already pointed out elsewhere (Proc. Roy. Soc. Edin., xxiii. p. 147), it is not necessary for the truth of only one of these identities that the whole of be axisymmetric, but merely a minor of it. For example, the first of the two here given, which manifestly may be written

holds if be axisymmetric. This explains the insertion of the theorem of § 17, which is really Kronecker's in an amended form.

page 527 note * So called perforce, as the less general expressions e.g. first brought to light by Kronecker already bear his name.

page 533 note * The proof desired at the end of § 10 will be found in a paper sent to the Edinburgh Math. Soc. on 29th Jan. 1902, and at present being printed as part of Vol. xx. of the Proceedings of that Society, under the title “The applicability of the Law of Extensible Minors to determinants of special form.”