Finite tensor arrays and the Frobenius inner product

Fix a field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, an integer $d\ge 2$, and positive integers $n_1,\dots ,n_d$. A finite order-$d$ tensor of shape $(n_1,\dots ,n_d)$ is an array $A=(A_{i_1,\dots ,i_d})$ indexed by $1\le i_k\le n_k$. Its Frobenius inner product and norm are $⟨A,B{⟩}_F={\sum }_{i_1=1}^{n_1}\cdots {\sum }_{i_d=1}^{n_d}\overline{A_{i_1,\dots ,i_d}}{B}_{i_1,\dots ,i_d},\mathrm{\parallel }A{\mathrm{\parallel }}_F=\sqrt{⟨A,A{⟩}_F}\mathrm{.}$ For a matrix, rank means the dimension of its column image and the Frobenius inner product is the same entrywise formula. The adjoint $M^{\ast }$ is conjugate transpose (transpose over $\mathbb{R}$).

For $1\le k\le d$, the mode-$k$ unfolding $A_{(k)}$ has $n_k$ rows and ${\prod }_{j\mathrm{\ne }k}{n}_j$ columns. Its row index is $i_k$ and its column index is the remaining tuple in increasing mode order, lexicographically ordered. Its entry at that row and column is the corresponding entry of $A$. For $1\le k<d$, the cut-$k$ unfolding $A^{⟨k⟩}$ has row tuple $(i_1,\dots ,i_k)$ and column tuple $(i_{k+1},\dots ,i_d)$, each lexicographically ordered. These conventions fix all reshapes used below.