  The perturbed GUE corners ensemble is the joint distribution of eigenvalues
of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where
$G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and
$\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov
transitions based on exponential jumps of eigenvalues, and show that their
successive application is equivalent in distribution to a deterministic shift
of the matrix. This result also leads to a new distributional symmetry for a
family of reflected Brownian motions with drifts coming from an arithmetic
progression.
  The construction we present may be viewed as a random matrix analogue of the
recent results of the first author and Axel Saenz (arXiv:1907.09155 [math.PR]).
