It is well known bow important positivity is in various branches of mathematics. For objects that are positive, one can usually obtain much more complete results than in the general case. For example, in linear algebra, positive and positive-definite matrices are among the most thoroughly studied matrix classes. It is equally important that the resulls of this study have been well documented: one can learn the properties of the matrix classes above from dozens of textbooks and monographs on matrix theory.
Things are quite different when from the global property of (positive or negative) definiteness one turns to the same property that holds only conditionally, i.e., as long as the argument does not leave a given subspace, orthant, polyhedron, or cone. We combine all these options under the terra "conditionally definite" matrices. Matrices of this kind are understood to a lesser extent than the classical positive-definite matrices. Moreover, to the best of our knowledge, no book exposition yet exists of the problem of conditional definiteness. At most, there are some survey papers devoted to the particular types of conditional-definite matrices.