Lewis Carroll identity
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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.
[edit] Formulation
Let A be an n × n matrix with entries in a commutative ring, and Aij (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
- (detA)(detB) = (detA11)(detA22) − (detA12)(detA21).
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.
[edit] 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
