Correlation of sign definiteness with reduction to perfect square of two quadratic forms bundle

The article discussed the interconnection between the sign definiteness bundle of two quadratic forms with simultaneous reduction of these forms to perfect squares by one linear real congruent transformation. The theorem about the necessary conditions for the sign definiteness of a bundle of two quadratic forms was ascertained. The obtained requirements fully coincide with the conditions for simultaneous diagonalization of matrices of these forms. The sufficient conditions for the sign definiteness of two quadratic forms are made up for the matrix forms simultaneously reduced to diagonal ones. Along with the known method of research the sign definiteness of two forms bundle an alternative approach was suggested. The approach makes it possible to make a parameter analysis of the considered problem. In the applied problems of qualitative analysis and the theory of stability of motion, the necessary conditions for the sign definiteness of a bundle of quadratic forms are applied at the initial stage of determining the stability area. Further, after the diagonalization of matrices, the sufficient conditions for stability are formed from the sufficient conditions for the sign definiteness of two quadratic forms bundle. The obtained results are demonstrated on the known solution of the stability problem of permanent rotation around the vertical axis of the Lagrange gyroscope. The system of analytical calculations on the advanced computers had been applied to carry out various computations related with expansion, substitution, factorization of expressions and calculation of matrix determinants.