[Authors] Filimonenkova, N. V. [Title] Sylvester’s criterion for m-positive matrices [AMS Subj-class] 15B48 Positive matrices and their generalizations; cones of matrices [Abstract] The paper is devoted to new concepts in algebra of symmetric matrices: the m-trace and the m-positive matrix. These concepts are derived from the development of the classical notions: the trace, the determinant of the matrix, the positive definite matrix, the m-th elementary symmetric function. The paper presents some known and new properties of m-positive matrices, including an analogue of Sylvester’s criterion. [Keywords] m-trace of matrix, m-positive matrix, Sylvester’s criterion [Comments] LaTeX, Russian, 12 pp. [Contact e-mail] nf33@yandex.ru