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Chiò’s and Dodgson’s determinantal identities. (English) Zbl 1288.15001

Summary: In this expository paper, we analyze and compare two determinantal identities constructed in the mid nineteenth century, F. Chiò’s [Mémoire sur les fonctions connues sous le noms de résultantes ou de déterminans (1853)] and C. L. Dodgson’s [The political pamphlets and letters of Charles Lutwidge Dodgson and related pieces. A mathematical approach. Compiled with introductory essays, notes, and annotations, by Francine F. Abeles (2001; Zbl 1036.01019)], from the perspective of their origins in earlier work by J. J. Sylvester [“On the relation between the minor determinants of linearly equivalent quadratic functions”, Philos. Mag., s. IV, vv. I, 295–305 (1851)] and C. G. J. Jacobi [J. Reine Angew. Math. 22, 285–318 (1841; Zbl 02751100)]. All the known proofs of Chiò’s and Dodgson’s identities are cited and we present and compare them as modern algorithms for evaluating determinants. Restated as a recurrence, we discuss the role of Dodgson’s identity in the development of the alternating sign matrix conjecture in the late twentieth century.

MSC:

15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15A15 Determinants, permanents, traces, other special matrix functions
01A55 History of mathematics in the 19th century

Biographic References:

Chiò, Felice; Dodgson, Charles Lutwidge

Software:

DODGSON; ROBBINS; rabbit
PDFBibTeX XMLCite
Full Text: DOI

References:

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