I'm trying to assess and prove whether a matrix containing only non-negative values (generated from the standard normal distribution) and diagonal values which are always positive (obtained as a product of the identity matrix and constants: $\alpha$ (size of the matrix)) and $\beta$ (a real number) is positive definite.
The matrix might be composed of two other matrices that easily create these results. For example: A multiplied by its transpose summed with the identity matrix.
$ AA' + \alpha\beta I $
This example would show the matrix to be symmetric, so we also need to show the positive definiteness
Is there some proof relating to the properties that allows a positive definite proof to be created without proving all eigenvalues to be positive for example?
EDIT: Originally forgot to specify that values of the matrix are from the standard normal distribution and there is a multiplier on the diagonal using the product of the length of the matrix and another real parameter