A matrix M is usually called a hollow matrix if all of its diagonal elements are zero:

$$ M_{pp} = 0, \quad \forall \: p. $$

We can generalize this to an $n$-way tensor T, such that:

$$ T_{p_1 \cdots p_n} = 0, \quad \mbox{if $p_i = p_j$ for some $i \ne j$}. $$

I call these types of tensors hollow. Are there other names for these?

Edit: made the question more precise.

  • 3
    $\begingroup$ Do you mean an element is zero if all indices $p_i$ are equal, or if at least two of them are equal? $\endgroup$ May 13, 2021 at 18:12
  • $\begingroup$ At least two of them are equal. $\endgroup$
    – twofiveone
    May 14, 2021 at 19:07
  • 1
    $\begingroup$ a special case, if it is also antisymmetric under exchange of any two indices, is called a completely antisymmetric tensor. $\endgroup$ May 14, 2021 at 20:27
  • $\begingroup$ This hollow condition is not invariant under conjugation by a linear transformation, so it doesn't seem very natural to me; it is basis dependent. It would be nice if you could include a sentence explaining where it might naturally arise, or why it is important. $\endgroup$
    – Ben McKay
    May 15, 2021 at 9:47
  • $\begingroup$ 1. It may occur in many-fermion wavefunctions due to the Pauli exclusion principle. Usually, it is also antisymmetric as @CarloBeenakker pointed out. $\endgroup$
    – twofiveone
    May 16, 2021 at 11:06


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