Symmetric Matrices Whose Entries Are Linear Functions

摘要

There exists a large set of real symmetric matrices whose entries are linear functions in several variables such that each matrix in this set is definite at some point, that is, the matrix is definite after substituting some numbers for variables. In particular, this property holds for almost all such matrices of order two with entries depending on two variables. The same property holds for almost all matrices of order two with entries depending on a larger number of variables when this number exceeds the order of the matrix. Some examples are discussed in detail. Some asymmetric matrices are also considered. In particular, for almost every matrix whose entries are linear functions in several variables, the determinant of the matrix is positive at some point and negative at another point.