# Norm of Tensor Product, Tensor Norm, Cubic Power and Gelfand Limit

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. 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. We show that if such a Gelfand limit exists for one norm, then it exists for all the other norms with the same value. This limit is zero for all third order nilpotent tensors.

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09/24/2019

### Norm of Tensor Product, Tensor Norm, and Cubic Power of A Third Order Tensor

We establish two inequalities for the nuclear norm and the spectral norm...
research
09/24/2019

### Tensor Norm, Cubic Power and Gelfand Limit

We establish two inequalities for the nuclear norm and the spectral norm...
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09/04/2019

### Spectral Norm and Nuclear Norm of a Third Order Tensor

The spectral norm and the nuclear norm of a third order tensor play an i...
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01/20/2020

### Nuclear Norm Under Tensor Kronecker Products

Derksen proved that the spectral norm is multiplicative with respect to ...
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07/07/2014

### Spectral norm of random tensors

We show that the spectral norm of a random n_1× n_2×...× n_K tensor (or ...
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04/12/2021

### Statistical inference of finite-rank tensors

We consider a general statistical inference model of finite-rank tensor ...
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07/09/2017

### On orthogonal tensors and best rank-one approximation ratio

As is well known, the smallest possible ratio between the spectral norm ...