  An $n \times n \times p$ tensor is called a T-square tensor. It arises from
many applications, such as the image feature extraction problem and the
multi-view clustering problem. We may symmetrize a T-square tensor to a
T-symmetric tensor. For each T-square tensor, we define a T-quadratic form,
whose variable is an $n \times p$ matrix, and whose value is a $p$-dimensional
vector. We define eigentuples and eigenmatrices for T-square tensors. We show
that a T-symmetric tensor has unique largest and smallest eigentuples, and a
T-quadratic form is positive semi-definite (definite) if and only if its
smallest eigentuple is nonnegative (positive). The relation between the
eigen-decomposition of T-symmetric tensors, and the TSVD of general third order
tensors are also studied.
