A method of computing the determinant of a square matrix due to Charles Dodgson (1866) (who is more famous under his pseudonym Lewis Carroll). The method is useful for hand calculations because, for an integer matrix, all entries in submatrices computed along the way must also be integers. The method is also implemented efficiently in a parallel computation. Condensation is also known as the method of contractants (Macmillan 1955, Lotkin 1959).

Given an n×n matrix, condensation successively computes an (n-1)×(n-1) matrix, an (n-2)×(n-2) matrix, etc., until arriving at a 1×1 matrix whose only entry ends up being the determinant of the original matrix. To compute the k×k matrix (n-1>=k>=1), take the k^2 2×2 connected subdeterminants of the (k+1)×(k+1) matrix and divide them by the k^2 central entries of the (k+2)×(k+2) matrix, with no divisions performed for k=n-1. The k×k matrices arrived at in this manner are the matrices of determinants of the k^2(n-k+1)×(n-k+1) connected submatrices of the original matrices.

For example, the first condensation of the 3×3 matrix

 [a b c; d e f; g h i]

yields the matrix

 [ae-bd bf-ce; dh-eg ei-fh],

and the second condensation yields


which is the determinant of the original matrix. Collecting terms gives


of which the nonzero terms correspond to the permutation matrices. In the 4×4 case, 24 nonzero terms are obtained together with 18 vanishing ones. These 42 terms correspond to the alternating sign matrices for which any -1s in a row or column must have a +1 "outside" it (i.e., all -1s are "bordered" by +1s).

See also

Alternating Sign Matrix, Chió Pivotal Condensation, Determinant, Determinant Expansion by Minors

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Bareiss, E. H. "Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination." Math. Comput. 22, 565-578, 1968.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.Dodgson, C. L. "Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values." Proc. Roy. Soc. Ser. A 15, 150-155, 1866.Lotkin, M. "Note on the Method of Contractants." Amer. Math. Soc. 55, 476-479, 1959.Macmillan, R. H. A New Method for the Numerical Evaluation of Determinants." J. Roy. Aeronaut. Soc. 59, 772, 1955.Robbins, D. P. and Rumsey, H. Jr. "Determinants and Alternating Sign Matrices." Adv. Math. 62, 169-184, 1986.

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Cite this as:

Weisstein, Eric W. "Condensation." From MathWorld--A Wolfram Web Resource.

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