  Pati (Physics Letters A, 2000) derived a sufficient condition for the
existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper,
we show that the condition is erroneous by demonstrating some counter-examples.
Moreover, we suitably modify the condition and provide a correctness proof. We
also show for the first time how this can be generalized to $n$-partite Hilbert
spaces for any $n\geq 2$. Finally, we prove that this condition is also a
necessary condition.
