  A polynomial ensemble is a probability density function for the position of
$n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det
\left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$.
Such ensembles appear frequently as the joint eigenvalue density of random
matrices. We present a number of transformations that preserve the structure of
a polynomial ensemble. These transformations include the restriction of a
Hermitian matrix by removing one row and one column, a rank-one modification of
a Hermitian matrix, and the extension of a Hermitian matrix by adding an extra
row and column with complex Gaussians.
