Positive definite matrix with an interesting estimate [duplicate]

Let $$A=(a_{ij}) \in \mathbb{R}^{n \times n}$$ be a symmetric matrix. For all $$i=1, \dots ,n$$ we have $$a_{ii} > \sum_{i \ne j} \vert{a_{ij}}\vert$$. I now have to show that $$A$$ is positive definite. I tried to look at the general case $$x^\top Ax$$, using the estimate above. Unfortunately without any luck... In the end I got something like $$x^\top Ax > \sum_{m=1}^n (x_m ^2 + x_m \min\{x_k : k \ne m\}) \sum_{m \ne k}a_{mk}$$.

I am unfortunately not allowed to use Gershgorins Circle Theorem as we did not discuss it up to this date :)

Any ideas to prove the statement?

~Cedric :)

marked as duplicate by Martin R, Jyrki Lahtonen, mechanodroid, Paul Frost, CesareoDec 2 '18 at 13:42

• Note that there is (at least) one answer which does not use Gershgorin's circle theorem, but does an explicit estimate on $x^T A x$. – Martin R Dec 2 '18 at 9:42
It's actually a simple estimation. We can see that $$\begin{eqnarray}x^T Ax &=&\sum_{1\leq i\leq n} a_{ii}x_i^2 + \sum_{1\leq i,j\leq n,i\neq j} a_{ij}x_ix_j\\&\geq &\sum_{1\leq i\leq n} a_{ii}x_i^2 - \sum_{1\leq i,j\leq n,i\neq j} |a_{ij}||x_i||x_j|\\ &\geq &\sum_{1\leq i\leq n} a_{ii}x_i^2 - \frac{1}{2}\sum_{1\leq i,j\leq n,i\neq j} |a_{ij}|(|x_i|^2+|x_j|^2)\\&=&\sum_{1\leq i\leq n} a_{ii}x_i^2 - \sum_{1\leq i,j\leq n,i\neq j} |a_{ij}||x_i|^2 \\&=&\sum_{1\leq i\leq n} (a_{ii}- \sum_{1\leq j\leq n,j\neq i} |a_{ij}|)|x_i|^2 >0, \end{eqnarray}$$ for all $$x = (x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\setminus\{0\}.$$ The fact that $$|a_{ij} | = |a_{ji}|$$ is used in the second equality.