ИСТИНА

There exists a number of papers devoted to the classification of smooth or ana- lytic matrix families. Such families naturally appear in the study of binary differential equations, dependency sets of vector fields on manifolds, as well as in connection with other problems in differential geometry. It is natural to consider such families up to G-equivalence, i. e., up to parameter-dependent linear base changes and parameter changes. In [1] analytic families of square matrices, which can be viewed as linear maps be- tween equidimensional spaces, are considered. In particular, normal forms of G-simple mappings (i. e., those having a finite number of adjacencies) are obtained. An ideologi- cally similar paper [2] is devoted to the study of analytic families of symmetric matrices. In [3] analytic families of skew-symmetric matrices are considered. In particular, a com- plete classification of two-parameter and a partial classification of three-parameter 4 × 4 simple skew-symmetric matrix families are obtained. We obtain a necessary existence condition for G-simple analytic symmetric and skew- symmetric matrix families that are even or odd in the totality of parameters in terms of number of parameters, matrix size and 1-jet rank. We also classify symmetric and skew-symmetric matrix families that are odd in the totality of parameters with 1-jet of corank zero. The work is generally inspired by aforementioned papers [1]–[3]. Some of the results presented can be found in [4]. The talk is based on our joint work with N. Abdrakhmanova and A. Terentiev. References [1] Bruce J. W., Tari F. On Families of Square Matrices // Cadernos de Mathematica. 2002. Vol. 3. P. 217–242. [2] Bruce J. W. On Families of Symmetric Matrices // Moscow mathematical journal. 2003. Vol. 3. P. 335–360. [3] Haslinger G. J. Families of Skew-symmetric Matrices. Ph. D. thesis. University of Liverpool. 2001. [4] Abdrakhmanova N. T., Astashov E. A. Simple germs of skew-symmetric matrix fam- ilies with oddness or evenness properties // Journal of Mathematical Sciences. 2023. Vol. 270, No. 5. P. 625-639.