Abstract
We formulate and prove necessary and sufficient conditions of simultaneous diagonalization of three real symmetric matrices of regular pencil. The conditions are algebraic and consist, in particular, of two spectral requirements and one matrix equality. For the degenerate matrix pencil we suggest an approach that allows reducing of 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 the stability of trivial solution to a system.
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