The invariance of property of principal leading minors’ of the square matrix positivity in the similarity transformation

The set of square matrices with positive principal leading minors and the similarity transformation of them using lower triangular matrices with positive diagonal elements are discussed. The criterion of the preservation of the principal corner minors' of a matrix positivity in the similarity transformation using the above-stated transformation is established. Moreover a sufficient condition of the preservation of the positive separation property for square above mentioned matrices in the similarity transformation is obtained. The received results are planned to be further used in the theory of controllability of asymptotic invariants of linear systems of ordinary differential equations.