I would like to update a covariance matrix $\mathbf{R}_T$ with a new incoming sample at time $T+1$, i.e. I would like a rank-1 update of the form $\frac{1}{T+1} [T \mathbf{R}_T + \mathbf{x}_{T+1}\mathbf{x}_{T+1}^{\top}]$. However I want a weighted average by forgetting past observations.
That is, I would like something of the form:
$$\mathbf{R}_{T+1}= \frac{1}{T+1} \big[\sum_{i=1}^{T-1} \alpha^{T-i}\mathbf{x}_{i}\mathbf{x}_{i}^{\top} + \alpha^0\mathbf{x}_{T+1}\mathbf{x}_{T+1}^{\top}\big] $$,
subject to $\sum \alpha=1$. But I would like to express $\mathbf{R}_{T+1}$ as function of $\mathbf{R}_{T}$, because I already have it. So I want to forget the data while retaining the covariance information. Could you please tell me where to search something related to it? I have read someting about the EWMA model, but not sure whether it's what I am searching.
Thanks.