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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!

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It's the symmetric version of the low-rank approximation problem.

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