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I am looking for help pointing me in the direction of any literature or other known work that analyze the probability distribution or other important properties of random variables of the form $AB^{-1}$, where $A,B$ are independent matrix random variables whose entries from some i.i.d. random variables, such as $N(0,1)$.

I am interested in seeing what techniques have been developed for analyzing such expressions, so that I can try to see if I can adapt any of those techniques to my own problem, which involves trying to obtain probabilistic bounds on a random variable that also has the form $AB^{-1}$.

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A study of the eigenvalues of $AB^{-1}$ would lead you to study solutions $\lambda$. of the equation ${\rm Det}\,(A-\lambda B)=0$. This "generalized eigenvalue problem" has been studied in some detail, see for example section 6 of How many eigenvalues of a random matrix are real?. The probability distribution of real solutions $\lambda$ is known exactly for independent standard normals $A,B$.

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  • $\begingroup$ Thank you, this is very interesting. I'm sorry that my question is rather vague. I know that the study of random matrices is often a messy business, so I just wanted to see what results were out there at all. $\endgroup$ May 28, 2014 at 0:47

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