Let $F\in \mathbb{R}^{n\times n}$ be an invertible matrix. I would like to better understand the relationship between $F$ and the vector $v\in \mathbb{R}^n$ satisfying
$ v = \text{argmin}_{z\in \mathbb{R}^n} \text{Var}(Fz) = \text{argmin}_{{z\in \mathbb{R}^n}\\ \|z\|_2=1} \|Fz-\bar{Fz}\|_2^2,$
where $\bar{Fz}$ is the $n\times1$ vector containing the average value of $Fz$ in each component. I would like to gain a better understanding of how $v$ and $F$ are related outside of the expression given above. Can $v$ be related to the singular value decomposition or eigendecomposition of $F$? Thank you.
Let $P$ denote the $n \times n$ matrix whose entries are all $1/n$. Note that we can write $$ \bar Fz = PFz. $$ As such, the vector $v$ satisfies $$ v = \operatorname{argmin}_{\|z\| = 1} \|Fz - PFz\| = \operatorname{argmin}_{\|z\| = 1} \|(I - P)Fz\|, $$ where $I$ denotes the identity matrix. Now, note that $I - P$ is a singular square matrix, which means that $(I - P)F$ is singular. Thus, the minimum must be attained with $(I - P)Fv = 0$.