Let $A$ be an $n\times n$ symmetric positive definite matrix with eigenvalues and eigenvectors $\lambda\_1\ge\lambda\_2\ge\cdots\ge\lambda\_n>0$ and $v\_1,v\_2,\cdots,v\_n$ respectively.


We know that the largest eigenvalue of $A$ can be obtained by "trace maximization" or "Rayleigh quotient maximization". I also noted that the largest eigenvalue of $A$ can be obtained via


$$\underset{x \in \mathbb{R}^n}{\text{minimize}} \quad \|A-xx'\|\_F^2 \tag{OP}$$


Specifically, $x = \pm \sqrt \lambda\_1 v\_1$ are minimizers of $\text{(OP)}$.


Is there a specific name for this optimization problem? Could you please direct me to some literature. Thanks!



Is there a specific name for this optimization problem?
It's the symmetric version of the [low-rank approximation](https://en.wikipedia.org/wiki/Low-rank_approximation) problem.


