On the optimal computation of a set of symmetric and persymmetric bilinear forms

A special class $\text{T}$ⁿ of n×n matrices is described, which has tensor rank n over the real field. A tensor base for general symmetric, persymmetric, both symmetric and persymmetric matrices and Toeplitz symmetric matrices can be defined in terms of the tensor bases of $\text{T}$^l for some different values of l. It is proved that both symmetric and persymmetric n×n matrices and Toeplitz symmetric n×n matrices have tensor rank $\text{(n+1)}{}^2\text{4}\text{]}$ and 2n?2, respectively, in the real field.