In my research, I have recently come across the following problem involving a recurrence formula, the problem statement:
We consider a given column vector $ x=(x_1,x_2,...,x_{d})^T \in \mathbb{R}^{d} $ and we define a column vector $ w = (w_1,w_2,...,w_{d})^T \in \mathbb{R}^{d} $ this denotes an initial condition of recurrence relation, and we define the following recursive relation $$w^{(n+1)} = w^{(n)}-\frac{2}{n}({w^{(n)}}^Tx-k)x$$ where $k$ is a nonnegative constant scalar, the initial condition is some arbitrary constant column vector $ w^{(1)} = (w_1,w_2,...,w_{d})^T=w \in \mathbb{R}^{d} $,
I was interested in finding a closed form for this recursive relation for $w^{(n)}$ for any $ n \in \mathbb{N} $ as a function of the index $n$ and the initial condition $w$, can someone please assist here? I thank all helpers.
$$\begin{array}{rl} \mathrm w_{k+1} &= \mathrm w_{k} - \dfrac{2}{k} \left( \mathrm w_{k}^{\top} \mathrm x - \gamma\right) \mathrm x\\\\ &= \mathrm w_{k} - \dfrac{2}{k} \mathrm x \left( \mathrm x^{\top} \mathrm w_{k} - \gamma\right)\\\\ &= \left( \mathrm I - \dfrac{2}{k} \mathrm x \mathrm x^{\top} \right) \mathrm w_{k} + \left(\dfrac{2 \gamma}{k}\right) \mathrm x\end{array}$$
Let
$$\mathrm A_k := \mathrm I - \dfrac{2}{k} \mathrm x \mathrm x^{\top} \qquad \qquad \qquad \mathrm b_k := \left(\dfrac{2 \gamma}{k}\right) \mathrm x$$
Hence,
$$\mathrm w_k = \mathrm A_{k-1} \cdots \mathrm A_2 \mathrm A_1 \mathrm w_1 + \left(\sum_{m=1}^{k-2} \mathrm A_{k-1} \mathrm A_{k-2} \cdots \mathrm A_{m+1} \mathrm b_m\right) + \mathrm b_{k-1}$$