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This question arose out of curiosity. Consider a positive definite hermitian matrix $A$ and a negative definite hermitian matrix $B$. Consider the product matrix $C=AB$. I read some where that positive definite matrices are some what like the positive real numbers and similarly negative definite matrices are like negative numbers. So does that kind of similarity carry over here, in detail, is $C$ negative definite?. A quick simulation tells me it is true for randomly generated examples, but I was looking for a mathematical proof if it is true.

EDIT--

I just figured out that this is equivalent to proving product of two positive definite matrices is positive definite.

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1 Answer 1

up vote 2 down vote accepted

$A=\begin{pmatrix}1 & 1\cr 1 & 4\end{pmatrix}$ $B=\begin{pmatrix}1 & 1\cr 1 & 5\end{pmatrix}$

the above two matrix is positive definte matrix, but $AB$ is not a symmetrical matrix. if we add $AB=BA$,then product of two positive definite matrices is positive definite.

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In essence, if the positive definite matrices commute, then their product is also positive definite. Its a beautiful result (though may be obvious) if you note that scalars also commute under multiplication. –  dineshdileep Oct 15 '12 at 15:14

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