Abstract: We perform a rigorous study of private matrix analysis when only the last $W$ updates to matrices are considered useful for analysis. We show the existing framework in the non-private setting is not robust to noise required for privacy. We then propose a framework robust to noise and use it to give first efficient $o(W)$ space differentially private algorithms for spectral approximation, principal component analysis (PCA), multi-response linear regression, sparse PCA, and non-negative PCA. Prior to our work, no such result was known for sparse and non-negative differentially private PCA even in the static data setting. We also give a lower bound to demonstrate the cost of privacy in the sliding window model.