## A multilinear singular value decomposition.(English)Zbl 0962.15005

This paper investigates higher order singular value decompositions (HOSVD) as multilinear generalization of singular value decomposition (SVD), in terms of higher order tensors exhibiting symmetries as motivated by applications in higher order statistics. This includes matrix representation of higher order tensors, their rank properties, generalization of scalar product, orthogonality, and Frobenius norm, and multiplication by a matrix. A HOSVD for real or complex $$N$$th order tensors is always possible and reduces to SVD when applied to a matrix. In psychometry it is called the Tucker model. The matrix of singular vectors can be computed from the sets of $$n$$-mode vectors in the same way as in the second order case.

### MSC:

 15A18 Eigenvalues, singular values, and eigenvectors 15A69 Multilinear algebra, tensor calculus 15A63 Quadratic and bilinear forms, inner products 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 62G30 Order statistics; empirical distribution functions 92B15 General biostatistics 62P15 Applications of statistics to psychology
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