Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Suppose we use the notation $a_{i}$ for the $i$-th row (or column, which would be the same given its symmetry), with $i\in\{1, ... , n\}$.
Suppose, in addition, that $A$ has the property that $\Vert a_i \Vert_2^2 < 1$, for every $i$.
I have the following queries:
- Can we say something about the eigenvalues of $A$?
- Ideally, can we link this property to $A$ being (or not) positive-definite?
Many thanks!