SUM OF MIXING DETERMINANTS

Abstract

Consider some square matrices of the same order. We can withdraw a given number of rows of the first matriz, a given number of rows of the second, etc., and form a new square matrix where the rows conserve their original ordinals. The sum of determinants of all possible miring matrices constitutes the introduced notion. The main theorem is that if the starting matrices are non-negative definite (n.n.d.) then the above sum is non-negative, a necessary and sufficient condition for its positiveness is given. Applications: measuring the steepness in multidimensional geometry, majorizing the error norm of least squares estimates.