As noted in 1806.05647, given a **symmetric matrix** $A$, the leading eigenvector value problem (LEVP)

$$Av = \lambda v,$$

where $A = A^T \in \mathbb{R}^{n \times n}$, $\lambda$ is the largest eigenvalue of $A$ and $v$ is the corresponding eigenvector, can be written as an unconstrained optimisation problem

$$\min_{x \in \mathbb{R}^n} f(x) \equiv \min_{x \in \mathbb{R}^n} \| A - xx^T\|^2_F,$$

where $\| \cdot \|_F$ denotes the Frobenius norm.

When the matrix $A$ is symmetric, the gradient of the function $f(x)$ is

$$\nabla f(x) = -4Ax + 4(x^Tx)x$$

and the optimal solution to the aforementioned optimisation problem can be shown to be $\pm \sqrt{\lambda}v$. All the coordinate-wise descent algorithms represented in the paper for computing the leading eigenvector depend on this result and the matrix $A$ being symmetric.

Is it possible to generalise this to the case when the matrix $A$ is not symmetric? In that case, the gradient is: $$\nabla f(x) = -2Ax -2A^Tx + 4(x^Tx)x$$

and the solution that was optimal for the symmetric matrix $A$ ( $\pm \sqrt{\lambda}v$) now is the optimal solution for $\frac{1}{2}(A^T + A)$ rather than $A$.