  Tubal scalars are usual vectors, and tubal matrices are matrices with every
element being a tubal scalar. Such a matrix is often recognized as a
third-order tensor. The product between tubal scalars, tubal vectors, and tubal
matrices can be done by the powerful t-product. In this paper, we define
nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and
establish several properties that are analogous to their matrix counterparts.
In particular, we introduce the irreducible tubal matrix, and provide two
equivalent characterizations. Then, the celebrated Perron-Frobenius theorem is
established on the nonnegative irreducible tubal matrices. We show that some
conclusions of the PF theorem for nonnegative irreducible matrices can be
generalized to the tubal matrix setting, while some are not. One reason is the
defined positivity here has a different meaning to its usual sense. For those
conclusions that can not be extended, weaker conclusions are proved. We also
show that, if the nonnegative irreducible tubal matrix contains a strongly
positive tubal scalar, then most conclusions of the matrix PF theorem hold.
