There exists a real number $c$ such that $A+cI$ is positive when $A$ is symmetric Without using the fact that symmetric matrices can be diagonalized: Let $A$ be a real symmetric matrix. Show that there exists a real number $c$ such that $A+cI$ is positive.
That is, if $A=(a_{ij})$, one has to show that there exists real $c$ that makes $\sum_i a_{ii}x_i^2 + 2\sum_{i<j}a_{ij}x_ix_j + c\sum_i x_i^2 > 0$  for any vector $X=(x_1,...,x_n)^T$.
This is an exercise in Lang's Linear Algebra.
Thank you for your suggestions and comments.

If $c$ is sufficiently big, you can complete the squares with the mixed terms and rewrite the left side as a sum of squares.
