I am given with a $N \times N$ matrix
$$ \begin{align*} A = \begin{bmatrix} 1 & \rho_{h} & \ldots & \rho_{h}^{N - 1} \\ \rho_{h} & 1 & \ldots & \rho_{h}^{N - 2} \\ \vdots & \vdots & \ddots & \vdots \\ \rho_{h}^{N - 1} & \rho_{h}^{N - 2} & \ldots & 1 \end{bmatrix}, \end{align*} $$
where $0 \leq \rho_h \leq 1$ and $N \geq 2$. I would like to approimate A with another $N \times N$ matrix given as:
$$ B= \begin{bmatrix} 1 & \rho & \ldots & \rho \\ \rho & 1 & \ldots & \rho \\ \vdots & \vdots & \ddots & \vdots \\ \rho & \rho & \ldots & 1 \end{bmatrix}. $$
This basically means, finding a relation between $\rho$ and $\rho_h$ such that $A \approx B$.
Any thoughts on, how can this be achieved?
You might want to use the least squares method, that is, to find a $\rho \in [0,1]$ that minimizes $$ \sum_{k=1}^N 2\sum_{j=1}^{k-1} (\rho - \rho_h^j)^2; $$ note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_{N,k}$, goes to $c_{k,k}$ and makes an edge there and continues to $c_{k,N}$, where $c_{k,k}$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $\| \cdot \|$ is a matrix norm (submultiplicative or not), you can think about minimizing $\|A - B\|$ as a function of $\rho$. In the above example, the norm was just the Euclidean norm on $\mathbb R^{n^2}$.