1 On the problem of diagonalizability of matrices over a principal ideal domain

By Volodymyr Prokip.

Let R – be a principal ideal domain with the unit element e ≠ 0 and U(R) the set of divisors of unit element e. Further, let R_{n} – the ring of (n × n)- matrices over R; I_{k} – the identity k × k matrix and O the zero n × n matrix. In this report we present conditions of diagonalizability of a matrix A ∈ R_{n}, i.e. when for A there exists a matrix T ∈ GL(n, R) such that T AT^{−1} – a diagonal matrix. Theorem. Let A ∈ R_{n} and

det(Ix − A) = (x − α_{1})^{k_{1}}(x − α_{2})^{k_{2}} ⋯ (x − α_{r})^{k_{r}},

where α_{i} ∈ R, and α_{i} − α_{j} ∈ U(R) for all i ≠ j. If m(x) = (x − α_{1})(x − α_{2}) ⋯ (x − α_{r}) – the minimal polynomial of the matrix A, i.e. m(A) = O, then for the matrix A there exists a matrix T ∈ GL(n, R) such that

T AT^{−1} = diag (α_{1}I_{k_{1}}, α_{2}I_{k_{2}}, …, α_{r}I_{k_{r}}) .