Approximate formulation of the probability that the Determinant or Permanent of a matrix un.pdf
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Approximate formulation of the probability that the Determinant
or Permanent of a matrix undergoes the least change under
perturbation of a single element
Genta Ito?
Maruo Lab., 500 El Camino Real #302, Burlingame, CA 94010, United States.
SUMMARY
In an earlier paper, we discussed the probability that the determinant of a matrix undergoes the least
change upon perturbation of one of its elements, provided that most or all of the elements of the
matrix are chosen at random and that the randomly chosen elements have a fixed probability of being
non-zero. In this paper, we derive approximate formulas for that probability by assuming that the
terms in the permanent of a matrix are independent of one another, and we apply that assumption
to several classes of matrices. In the course of deriving those formulas, we identified several integer
sequences that are not listed on Sloane’s Web site. Preprint submitted to Numer. Linear Algebra
Appl. on June 28, 2007
Prepared using nlaauth.cls
key words: Determinant; Permanent; Permutation; Sloane’s sequences
1. Introduction
In an earlier paper [1], we discussed the problem of finding the probability that the determinant
of a matrix undergoes the least change under perturbation of one of its elements. In this paper,
we consider only the case where the randomly chosen matrix elements are values of a continuous
(real) random variable, and we derive approximate formulas for that probability by assuming
that the terms in the permanent of a matrix are mutually independent. Denote the determinant
of any matrix A by detA. For an (n+ 1)× (n + 1) matrix Mn+1, the expansion of detMn+1
via row i is
detMn+1 = mi1Mi1 + · · ·+mijMij + · · ·+min+1Min+1, (1)
where mij is the element of Mn+1 at the intersection of row i and column j, and Mij is the
cofactor of mij . Mij can be written as (?1)
i+j detSn, where Sn is the n × n submatrix of
Mn+1 which is obtained by deleting row i
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