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On some properties of three different types of triangular blocked tensors

We define three types of upper (and lower) triangular blocked tensors, which are all generalizations of the triangular blocked matrices. We study some basic properties and characterizations of these three types of triangular blocked tensors. We obtain the formulas for the determinants, characteristic polynomials and spectra of the first and second type triangular blocked tensors, and give an example to show that these formulas no longer hold for the third type triangular blocked tensors. We prove that the product of any two $(n_1,\cdots,n_r)$-upper (or lower) triangular blocked tensors of the first or second or third type is still an $(n_1,\cdots,n_r)$-upper (or lower) triangular blocked tensor of the same type. We also prove that, if an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the first or second or third type has a left $k$-inverse, then its unique left $k$-inverse is still an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the same type. Also if it has a right $k$-inverse, then all of its right $k$-inverses are still $(n_1,\cdots,n_r)$-upper triangular blocked tensors of the same type. By showing that the left $k$-inverse (if any) of a weakly irreducible nonsingular $M$-tensor is a positive tensor, we show that the left $k$-inverse (if any) of a first or second or third type canonical $(n_1,\cdots,n_r)$-upper triangular blocked nonsingular $M$-tensor is an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the same type all of whose diagonal blocks are positive tensors. We also show that every order $m$ dimension $n$ tensor is permutation similar to some third type normal upper triangular blocked tensor (all of whose diagonal blocks are irreducible). We give an example to show that this is not true for the first type canonical upper triangular blocked tensor.