TENSOR RANK, SIMULTANEOUSLY DIAGONALIZATION, AND SOME RELATED MATRIX VARIETIES

In this article, I will first give a criterion for a generic m × n × n tensor to have rank n using simultaneously diagonalization of certain set of matrices. Then, in Section 3, I will study some properties (like irreducibility and diemnsion) of the varieties defined by the above si-multaneous diagonalization condition which are closely related to the commuting varieties. This criterion also provides a possible way to attack the ”Salmon Conjecture”. 1