Signdefiniteness and reduction to full squares for the bundle of tree quadratic forms

The paper discusses the investigation of the relation of signdefiniteness of the bundle of three quadratic forms with to the simultaneous diagonalization by congruent transformation of matrices of the respective quadratic forms. sufficient conditions for the non-definiteness of the bundle of three quadratic forms have been obtained. These conditions are bound up with the impossibility of simultaneous diagonalization of any two matrices of these forms and satisfaction of one matrix inequality. In case of simultaneous diagonalization of any two matrices and satisfaction of the latter matrix equality, simultaneous diagonalization of all three matrices is possible. Signdefinite and signvariable bundles of three quadratic forms are shown in the case when the last matrix equality fails to be satisfied. Signdefiniteness of the bundle of three quadratic forms reduced to full squares is considered. For the purpose of investigation of signdefiniteness of such bundles of forms we have proposed an alternative approach based om analysis of quadratic forms of four variables. The issue of signconstancy of bundles of three signvariable quadratic forms has been investigated. Demonstrative examples are given.