On one class of positive matrices with minors of different signs

A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named strictly $\mathcal J$-sign-symmetric. The existence of the second (according to the module) positive simple eigenvalue $\lambda_2$ of a positive matrix $A$ is proved under the additional condition, that its compound matrix belongs to the introduced class of strictly $\mathcal J$-sign-symmetric matrices.