Simultaneous diagonalization of three real symmetric matrices

We formulate and prove necessary and sufficient conditions of simultaneous diagonalization of three real symmetric matrices of regular pencil to diagonal ones. The conditions are algebraic and consist, in particular, of two spectral requirements and one matrix equality. For degenerate matrix pencil we suggest an approach that allows to reduce the analysis to a regular pencil. With the use of obtained theorems we investigate a decomposition of linear gyroscopic system into subsystems of an order not higher than two and a stability of trivial solution to a system.