Norm of Tensor Product, Tensor Norm, and Cubic Power of A Third Order Tensor
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We establish two inequalities for the nuclear norm and the spectral norm of tensor products. The first inequality indicts that the nuclear norm of the square matrix is a matrix norm. We extend the concept of matrix norm to tensor norm. A real function defined for all real tensors is called a tensor norm if it is a norm for any tensor space with fixed dimensions, and the norm of the tensor product of two tensors is always not greater than the product of the norms of these two tensors. We show that the 1-norm, the Frobenius norm and the nuclear norm of tensors are tensor norms but the infinity norm and the spectral norm of tensors are not tensor norms. We introduce the cubic power for a general third order tensor, and show that the cubic power of a general third order tensor tends to zero as the power increases to infinity, if there is a tensor norm such that the tensor norm of that third order tensor is less than one. Then we raise a question on a possible Gelfand formula for a general third order tensor. Preliminary numerical results show that a spectral radius-like limit exists in general.
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