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


No, this won't necessarily happen, for fairly basic reasons: for example, $$ \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} $$ has negative determinant, so it can't be positive-definite.

  • $\begingroup$ Thanks @Chappers, I realised from this basic example I might need to update my question as the identity matrix is multiplied by a constant relating to the size of the matrix and the rest of the values are generated from the standard normal distribution! $\endgroup$ – jfive Jan 7 '16 at 21:11
  • $\begingroup$ Have you met Sylvester's Criterion? That's one simple way of determining if a Hermitian matrix is positive-definite without going all the way to the eigenvalues. $\endgroup$ – Chappers Jan 7 '16 at 21:16
  • $\begingroup$ I have updated the question with the additional information, perhaps now I can use this property I found elsewhere somehow: Namely, AA is positive-definite if all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column. See ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/… $\endgroup$ – jfive Jan 7 '16 at 21:21

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