# Product of a diagonal matrix with all positive entries and a quadratic matrix

I have a diagonal matrix D with all positive entries on diagonal and a quadratic matrix $$A=B^{T}B$$, is the product of the two matrices $$C=DA$$ also positive definite? or does the product has other properties? Thanks!

• why don't you try some very simple ($2\times 2$) examples for $D$ when $A=\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}$ (i.e. the all ones matrix) Feb 14, 2021 at 18:45
• $A$ considered here is positive definite, is the result also positive definite and I'm wondering is there any simple proof for that? Thanks! Feb 14, 2021 at 20:25

The product doesn't have to be positive definite. For a simple counterexample take any positive semidefinite $$A$$ (any positive semidefinite $$A$$ can be written as $$B^T B$$). Then take $$D$$ to be the identity matrix and $$C = DA = IA = A$$ is not positive definite.
Edit The product need not be positive semidefinite either. For the product to be positive semidefinite one would require $$DA$$ to be a symmetric matrix. However, this is the case iff $$A$$ and $$D$$ commute, i.e. $$[A,D]=0$$. Which will not be the case for most choices of $$A$$ and $$D$$. You will however still have that the eigenvalues of $$DA$$ are all non-negative.
• Thanks so much for the updated detailed answer. Yes, I can draw that the eigenvalues of the product are all non-negative. Then, I want to ask for this case can I say $u^{T} DA u \geq 0$? Or if I give the additional condition that $B$ is of full rank, can I say that the resultant product is positive definite? Feb 24, 2021 at 8:34
• How about the matrix form is $$BAB^{T}$$ where A is the diagonal matrix with all positive entries? Thanks! May 2, 2021 at 6:05