**4. Conclusion**

Because of the great practical application eigenvalue problem occupies an important place in linear algebra. In this chapter, we discussed the linear and quadratic eigenvalues. In particular, an emphasis is on numerical methods such as the QR algorithm, Rayleigh quotient iteration for linear problems of eigenvalues and linearization and minmax characterization of quadratic problems eigenvalues. The whole chapter shows that the structure of the matrix, participating in the problem of eigenvalues, strongly influence the choice of the method itself. It is also clear that using the features of the structure matrix can do much more effectively existing algo‐ rithms. Thus, further studies are going to increase of use feature matrix involved in the problem of eigenvalues, with the aim of improving the effectiveness of the method. Finally, we point out that in this chapter we introduced new Rayleigh functionals for gyroscopically stabilized system that enables complete minmax (maxmin) characterization of eigenvalues. It's a new strategy. We have proved all relevant features of new Rayleigh functionals.
