To the problem of solution symmetry for linear matrix differential equations

The symmetry of Cauchy problem solution for linear matrix differential equations is under research in the present article. The coefficients of the equation in question are supposed to be analytical functions in some domain of the complex plane. We find a formula for high-order derivatives of an arbitrary solution of the equation. We prove the sufficient conditions for the symmetry of Cauchy problem solution for linear matrix differential equations on the basis of the devised formula. To check these conditions, we need to analyse the properties of the special matrix sequence. Since the sequence consists of the infinite number of elements, the check is difficult to implement. It is shown that if some requirements are met, then it is sufficient to check only first several elements of the sequence. The example of the linear matrix differential equation is given to illustrate how the proposed condition may be used in proving the solution symmetry. The obtained results may be used in solving various problems of the control theory.

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