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Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist $u_j,v_j,c_j\in R^n$ with $$R(x)=\sum_{j=1}^N (c_j^Tx)u_jv_j^T$$ for all row vectors $x$ with $n$ entries.

We have $N(n)\le n(n+1)/2$ since $R=\sum_{i\le k} R_{ik}e_ie_k^T$, where $e_i$ is the $i$th unit vector. It is not difficult to see that this bound is sharp for $n\le 2$. I am interested in good lower and upper bounds for other small $n$ and for large $n$.

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  • $\begingroup$ You might consider adding a top-level / arXiv tag in order to increase the visibility of your question. $\endgroup$
    – Stefan Kohl
    Oct 11, 2015 at 11:02

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