On Partly and Nearly Decomposable Tensors

In this paper, we extend the theory of non-negative tensors.We present a condition that ensures the existence of at least k zero entries on every diagonal of a non-negative tensor. We define partly decomposable and fully indecomposable tensors. Furthermore, we present a necessary and sufficient condition for a tensor to be partly decomposable. We study the positivity of the permanents of fully indecomposable tensors and the positivity of the permanents of subtensors of dimension n − 1 of fully indecomposable tensors.We obtain a relation between non-negative, partly decomposable and non-negative, reducible tensors. We find a sufficient condition for block tensors to be fully indecomposable. We define nearly reducible and nearly decomposable tensors. Also, we present a sufficient condition for the main diagonal blocks of a block tensor to be nearly decomposable. We present a simple canonical form for nearly reducible tensors.