Invariance and commutativity properties of some classes of solutions of the matrix differential equation X(t)Xʹ(t) = Xʹ(t)X(t) Original Research Article

This paper presents some new results about the 47 years old study of the matrix differential equation XXʹ = XʹX. First, we prove that if for every t in an interval Ω subset of or equal to image a solution X(t) set membership, variant imagen × n is similar to diag[Jq(0), …, Jq(0)], then for every k set membership, variant {0, …, q} the kernel and the image of X(t)k do not depend on t set membership, variant Ω. Furthermore, we show that in this case, the obvious nilpotency of index q of X(t), imageX(t)q = 0 for allt set membership, variant ω spreads on Ω: imageX(t1)cdots, three dots, centeredX(tq) = 0n for allt1,…,tq set membership, variant ω This implies that when q = 2, the solution X is necessarily commutative: X(s)X(t) = 0n = X(t)X(s) for every s, t set membership, variant Ω. However, we show that when q = 3, there are analytic solutions that are not commutative. Also, we completely describe the solutions of XXʹ = XʹX such that X2 = 0. Finally, we construct a class of analytic solutions whose kernel and image are not constant.