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We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or n-way arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor extension of the Eckart--Young SVD approximation theorem by Leibovici and Sabatier Linear Algebra Appl., 269 (1998), pp. 307--329.
Tamara G. Kolda (Mon,) studied this question.