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 semidefinite matrices and diagonalization?
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3I think this wiki article should do the deal: http://en.wikipedia.org/wiki/Square_root_of_a_matrix – Dec 21 '11 at 21:21

Related posts: http://math.stackexchange.com/questions/57292/foreverymatrixainm2mathbbctheresxinm2mathbbcs, http://math.stackexchange.com/questions/65227/squarerootofamatrix, http://math.stackexchange.com/questions/72551/aquestionaboutntimesnmatrix – Jonas Meyer Dec 21 '11 at 21:36
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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.
Mikko Korhonen
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4"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