Lewis Carroll identity

The Lewis Carroll identity is an identity involving minors of a square matrix proved by Charles Dodgson (pseudonym Lewis Carroll), who used it in a method of numerical evaluation of matrix determinants called the Dodgson condensation. From the modern perspective, the Lewis Carroll identity expresses a "straightening law" in the algebra of polynomial functions of matrices.

Formulation

Let "A" be an "n" × "n" matrix with entries in a commutative ring, and "A"_"ij" ("i", "j" = 1, 2) denote its ("n" − 1) × ("n" − 1) submatrices of "A" formed by the ("n" − 1) first ("i" = 1) or last ("i" = 2) rows, and the ("n" − 1) first ("j" = 1) or last ("j" = 2) columns. Let "B" be the ("n" − 2) × ("n" − 2) submatrix of "A" formed by the rows and columns from 2 to "n" − 1. The Lewis Carroll identity states that

: $(det\; A)(det\; B)=(det\; A\_\{11\})(det\; A\_\{22\})\; -\; (det\; A\_\{12\})(det\; A\_\{21\}).$

If the matrix "B" is nonsingular, then dividing by its determinant leads to an expression for the determinant of "A" in terms of the determinants of orders 1 and 2 lower than the order of "A". Recursive application of this procedure is the method of Dodgson condensation.

References

* David Bressoud, "Proofs and Confirmations", MAA Spectrum, Mathematical Association of America, Washington, D.C., 1999

* C. Dodgson, "Condensation of determinants", Roy. Soc. London Proc. 15 (1866), 50–55