We investigate the intertwining of Laguerre processes of parameter \(\alpha \) in different dimensions. We introduce a Feller kernel that depends on \(\alpha \) and intertwines the \(\alpha \) -Laguerre process in \(N+1\) dimensions and that in N dimensions. When \(\alpha \) is a non-negative integer, the new kernel is interpreted in terms of the conditional distribution of the squared singular values: if the singular values of a unitarily invariant random matrix of order \((N+\alpha +1) \times (N+1)\) are fixed, then the those of its \((N+\alpha ) \times N \) truncation matrix are given by the new kernel.
