On the uniqueness of multilinear decomposition of N-way arrays

INTRODUCTION Consider an I # J matrix X and suppose that rank (X) = 3. Let x i,j denote the (i, j)th entry of X.Thenit holds that x i,j admits a three-component bilinear decomposition x i#j # # 3 f #1 a i#f b j#f #1# for all i = 1,...,I and j = 1,...,J. Equivalently, letting a f := [a 1,f ,...,a I,f ] T and similarly for b f , X # a 1 b T 1 # a 2 b T 2 # a 3 b T 3 #2# i