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More specifically, how do you define the square root of an $n\times n$ matrix A and express it in linear algebra terms? Does this have something to do with positive semi-definite matrices and diagonalization?

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Square root of a matrix $A$ is another matrix $B$ such that $B^2 = A$. It might or might not exist and it might or might not be unique. See Wikipedia for more.

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"might or might not be unique": It is never unique unless $n=1$ and $A=0$, or unless you impose additional conditions such as positivity. (If $A$ is positive semidefinite, then it has a unique positive semidefinite square root.) – Jonas Meyer Dec 21 '11 at 21:29

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