Abstract: We show that given an estimate $\widehat{\text{mat}A}$ that is close to a general high-rank positive semi-definite (PSD) matrix $\text{mat}A$ in spectral norm (i.e., $\Vert \widehat{\text{mat}A}-\text{mat}A{\Vert }_2\le \delta$), the simple truncated Singular Value Decomposition of $\widehat{\text{mat}A}$ produces a multiplicative approximation of $\text{mat}A$ in Frobenius norm. This observation leads to many interesting results on general high-rank matrix estimation problems: 1.High-rank matrix completion: we show that it is possible to recover a {general high-rank matrix} $\text{mat}A$ up to $(1+\epsilon )$ relative error in Frobenius norm from partial observations, with sample complexity independent of the spectral gap of $\text{mat}A$. 2.High-rank matrix denoising: we design algorithms that recovers a matrix $\text{mat}A$ with relative error in Frobenius norm from its noise-perturbed observations, without assuming $\text{mat}A$ is exactly low-rank. 3.Low-dimensional estimation of high-dimensional covariance: given $N$ i.i.d.~samples of dimension $n$ from ${\mathcal{N}}_n(\text{mat}0,\text{mat}A)$, we show that it is possible to estimate the covariance matrix $\text{mat}A$ with relative error in Frobenius norm with $N\approx n$,improving over classical covariance estimation results which requires $N\approx n^2$.