Let $k,n$ be fixed natural numbers and $a_{j_1,\ldots,j_k}$ be complex numbers with $\sum\limits_{1\le j_1,\ldots,j_k\le n}|a_{j_1,\ldots,j_k}|^2=1$.
Observe that if we have a $n\times n$ matrix $(z_{ij})$ such that each row is unit vector i.e. $\sum_j|z_{ij}|^2=1$, then for any $1\le i_1,\ldots,i_k\le n$, we have $\sum\limits_{1\le j_1,\ldots,j_k\le n}|z_{i_1j_1}\ldots z_{i_kj_k}|^2=1$.
BY HOSVD, it can be guaranteed that $a_{j_1,\ldots,j_k}=\sum\limits_{1\le\alpha_1,\ldots,\alpha_k\le n}s_{\alpha_1,\ldots,\alpha_k}(U_1)_{\alpha_1j_1}\ldots (U_k)_{\alpha_k j_k}$ where $U_1,\ldots,U_k$ are unitary matrices and $\sum\limits_{\alpha_1,\ldots,\alpha_k}|s_{\alpha_1,\ldots,\alpha_k}|^2=1$.
I can define $k\times n$ matrix $Y=(y_{ij})$ by $y_{ij}=(U_i)_{\alpha_i j}$.Then$a_{j_1,\ldots,j_k}=\sum\limits_{1\le\alpha_1,\ldots,\alpha_k\le n}s_{\alpha_1,\ldots,\alpha_k}Y_{1j_1}\ldots Y_{kj_k}$.
Now, this Y is a $k\times n$ matrix. If $k$ is larger than n, then it is not desirable. I want to get a $n\times n$ matrix (or a m\times n where m does not depend upon m) $Z$ such that these $a_{j_1,\ldots,j_k}$'s can be approximated "somehow" by the elements of the form $z_{i_1j_1}\ldots z_{i_kj_k}$. I am curious is this possible or not.
Can anyone help me in this regard? Thanks for your help in advance.