Hi Everyone,


Assume that we have a real symmetric matrix $H$, which can be written in the form $H=D \cdot B$, where $D$ is a positive diagonal matrix, and $B$ is a diagonally dominant matrix.
All elements of all matrices are positive real numbers.
We know that real symmetric matrices have real eigenvalues, and that diagonally dominant matrices have (potentially complex) eigenvalues with positive real parts. Could we infer from the above that $H$ is positive definite?


More generally, if $D$ is a diagonal matrix and $B$ is a positive definite matrix, could we infer that the product $D \cdot B$ is positive definite?
My feeling is that this problem should have long been solved, I would really appreciate any pointers to books/research articles that talk about this problem. 
Thanks! :-)



Eigenvalues of Matrix Product
The answer is **Yes**. Write $B=D^{-1}H$. Thus $B$ is the product of two Hermitian matrices, ones of which ($D$) being positive definite. It is a classical fact (see my book on Matrices, 2nd edition, Prop. 6.1) that this product is diagonalisable with real eigenvalues of the same signs as those of $H$. The $B$ has real eigenvalues, which are positive because you already knew that their real parts are positive. This implies that the eigenvalues of $H$ are positive. Hence $H$ is positive definite.


