APPENDIX C
VANDERMONDE DETERMINANTS, SYMMETRIC AND ALTERNATING
POLYNOMIALS
C.1 Introduction
An order n determinant whose i,j element is given by (j(xi) for some set of n scalar
functions j and a companion set of n scalar variables xi, i.e.
V(x.) 2(x.) *... n(xi)
(X2) 02(X2) ... On(X2)
A,(x)= Eq. C.1
(X,) 2(Xn) "' Xn)
is called an alternant. The name derives from the fact that exchanging any pair of the
variables in its argument list (e.g. x, and Xq) affects the value of An(x)= A,(x,,, x, *, x,)
only by reversing its algebraic sign. This property is clear from Eq. C.1, because trans-
posing the variables Xp and Xq in An(x) amounts to exchanging the corresponding rows of
the determinant, and from an elementary property of determinants, any such row
exchange leaves the determinant value unchanged in magnitude but reversed in sign.
The state transition matrix of the Markov chain representing the simple genetic
algorithm, as introduced in Section 4 of this paper, is a multivariate generalization of the
matrix form underlying the alternant. The coefficient symmetries noted in Section 9.5 in
connection with the stationary distribution representation development are a conse-
quence. Section 10 proposes exploiting this connection in continuing the stationary distri-
bution representation work begun in Section 9. This appendix provides some of the
related background.