How to replace a PSD matrix with a smaller PSD matrix?

Question

Suppose $\mathbf{x}$ is a d-dimensional vector where $l \le \|\mathbf{x}\|\le L$. We define $\mathbf{V}= \mathbf{x} \mathbf{x}^\top$ which is a PSD matrix with eigenvalues 0 and $\|\mathbf{x}\|^2$. I am wondering if it is possible to find a matrix in form of $a\mathbf{I}$ ($\mathbf{I}$ is Identity matrix) with $a>0$ such that $\mathbf{V} \succeq a \mathbf{I}$ (or equivalently $\mathbf{V} - a\mathbf{I}$ is a PSD matrix). If it is not possible, should I add any condition like the magnitude of elements of vector $\mathbf{x}$,e.g., $|x_i|\ge b>0$, to find this matrix?