In the paper, we address the important problem of tensor decomposition which can be seen as a generalisation of Sin- gular Value Decomposition for matrices. We consider gen- eral multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and we give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algo- rithm is described: it applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations in- volved in this approach and how the decomposition can be recovered from eigenvector computation.