| Author(s) | Collection number | Pages | Download abstract | Download full text |
|---|---|---|---|---|
| Koliada R. V., Melnyk O. M., Prokip V. M. | № 2 (59) | 56-64 |
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It is said that an matrix is the square root of an matrix over a field if and only if . The square roots of matrices are the main concepts for calculation of the polar decomposition of a matrix, solvability of differential equations and for construction of Markov financial model. It is evident that a matrix has a square root if and only if the matrix equation is solvable. It is known that square roots of complex numbers always exist, but square roots of matrices over the field of complex numbers may not exist. On the other hand, a non-zero matrix can have an infinite number of square roots. The problem of existence of square roots for matrices over the field of complex numbers is well studied. It may be noted that the problem of existence of square roots of matrices over an arbitrary field has been poorly researched.
In this article we study the problem of existence of square roots of matrices over an arbitrary field . Under certain restrictions we give necessary and sufficient conditions for existing the square root with given characteristic polynomial for a matrix over a field . It has been proved if the given square root exists, then it is uniquely defined by polynomial and we suggest a method of finding it. This allows us to give a complete solution to the problem of square roots of matrices over arbitrary field with relatively prime elementary divisors.
This research method allows us to specify the classes of matrices for which the number of square roots is finite.
Keywords: Square root of a matrix, rank of matrix, characteristic polynomial, linear matrix equation.
doi: 10.32403/1998-6912-2019-2-59-56-64
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