Abstract

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.

中文翻译：

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输入为线性函数的对称矩阵

摘要

存在大量的实对称矩阵，其条目是几个变量中的线性函数，因此该集合中的每个矩阵在某个点上都是确定的，也就是说，在用一些数字代替变量之后，矩阵是确定的。特别是，此属性几乎适用于所有此类的二阶矩阵，其条目取决于两个变量。当该数量超过矩阵的阶数时，几乎所有具有其项的变量都取决于条目的几乎所有二阶矩阵，该属性均适用。详细讨论了一些示例。还考虑了一些非对称矩阵。特别是，对于几乎每个条目都是在多个变量中具有线性函数的矩阵，矩阵的行列式在某个点上为正，而在另一点上为负。