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It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems. It is shown in this paper that a complex symmetric matrix can be diagonalised by a (complex) orthogonal transformation, when and only when each eigenspace of the matrix has an orthonormal basis; this implies that no eigenvectors of zero Euclidean length need be included in the basis. If the matrix cannot be diagonalised, then it has at least one invariant subspace which consists entirely of vectors of zero Euclidean length.

For both differentiable and nondifferentiable functions defined in abstract spaces we characterize the generalized convex property, here called cone-invexity, in terms of Lagrange multipliers. Several classes of such functions are given. In addition an extended Kuhn-Tucker type optimality condition and a duality result are obtained for quasidifferentiable programming problems.

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A unified treatment is given of generalizations of Farkas' theorem on linear inequalities to arbitrary convex cones and to dual pairs of real vector spaces of arbitrary dimension. Various theorems for locally convex spaces readily follow. The results are applied to duality and converse duality theory for linear programming and to a generalization of the Kuhn-Tucker theorem, both of these in spaces of arbitrary dimension and with inequalities involving arbitrary convex cones.1. Introduction. Farkas' theorem on systems of linear inequalities 11] (see also [ 10, Thm. 4-1) has been generalized to systems involving polyhedral cones [ 1], and to arbitrary cones in locally convex spaces [12]. A unified theory is presented here, based on arbitrary cones and dual pairs of real vector spaces. A necessary and sufficient condition is obtained (Theorem 2) for the solvability of a linear equation Ax b by a vector x in a given convex cone. A related necessary and sufficient condition (Theorem 1) is obtained as an inclusion relation between two cones. Although these results are closely related to the Hahn-Banach separation