On a dynamical system on matrix algebra (Q1340468)

It is considered a dynamical system defined by the nonlinear ordinary differential equation \({dX \over dt} = [[X^*,X],X]\), where \([X,Y] = XY - YX\), \(X\) and \(Y\) are complex \(n \times n\) matrices, \(X^*\) is the Hermitian adjoint of \(X\). This system arises as the gradient flow for two kinds of variational problems on the matrix algebra. The properties of this dynamical system are explored. It is proved that for any \(X_ 0\) the trajectory \(X(t)\) of the dynamical system starting at \(X_ 0\) exists for all \(t \geq 0\) and always lies in the conjugacy class of \(X_ 0\). Furthermore \(t \| [X^*(t), X(t)] \|^ 2\) tends to 0 as \(t \to \infty\), where \(\| X \|\) is a certain norm of matrix \(X\). This means that the \(\omega\)-limit set of the trajectory is in the set of normal matrices that is matrices commuting with their Hermitian adjoints. One of the main results is the following. Theorem. If the initial matrix \(X_ 0\) is semisimple, then the trajectory \(X(t)\) converges exponentially to a normal matrix \(X_ \infty\) which is conjugate to \(X_ 0\) as \(t \to \infty\). In particular, \([X^*(t), X(t)]\) converges to 0 exponentially as \(t \to \infty\). If \(X_ 0\) is not semisimple, then the behavior of the trajectory is completely different in the sense that \[ \int_ 0^ \infty t \bigl \| [X^*(t), X(t)] \bigr \|^ 2; \quad dt = \infty. \] The rate of exponential decay is given explicitly in terms of the eigenvalues of \(X_ 0\).