  Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let
$\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the
zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with
$K\ge N$, the $N$ eigenvalues of
$\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$
form a Markov process with an explicit transition kernel. This generalizes a
classical result of Rogers and Pitman for multidimensional Brownian motion with
drift which corresponds to $N=1$. We then give two more descriptions for this
Markov process. First, as independent squared Bessel diffusion processes in the
wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned
to never intersect. Second, as the distribution of the top row of interacting
squared Bessel type diffusions in some interlacting array. The last two
descriptions also extend to a general class of one-dimensional diffusions.
