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Let $A\in\mathbb{R}^{n\times n}$ be a symmetric nonsingular positive definite matrix, and $B\in\mathbb{R}^{n\times n}$ be a symmetric singular positive semi-definite matrix, this is

$$\begin{array}{l} A = U{\Lambda _1}{U^{ - 1}}\\ B = Q{\Lambda _2}{Q^{ - 1}} \end{array}$$

where the matrices $U,Q\in\mathbb{R}^{n\times n}$ are orthogonal and $\Lambda _1=diag\left\{ {{\lambda _1}, \cdots {\lambda _n}} \right\}$, $\Lambda _2=diag\left\{0, {{\gamma _2}, \cdots {\gamma _n}} \right\}$ are diagonal where their entries are real and positive eigenvalues. Then what can I say about $C=AB$, how are related the eigenvalues of $A$ and $B$ with the ones on $C$.

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Related : math.stackexchange.com/questions/143362/… –  Ewan Delanoy Nov 4 '13 at 11:26
    
For starters: $det(A*B)=det(A)*det(B)$, we also know that $det(A)= \prod_{i=1}^n \lambda_{ai}$ and $det(B)= \prod_{i=1}^n \lambda_{bi}$. So $det(A*B)= \prod_{i=1}^n \lambda_{ai} \lambda_{bi}$. But I guess there is a lot more to discover... –  Leo Nov 4 '13 at 11:43
    
Let $A,B$ be symmetric real matrices. If $A>0$ then $AB$ and $B$ have same signature. –  loup blanc Nov 18 '13 at 18:10

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