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Square root of a matrix
Prove that if all eigenvalues of a matrix $A \in \mathcal{M} (n,n; \mathbb{R} )$ are real, then there exists $B \in \mathcal{M} (n,n; \mathbb{R} ) $ such that $B^2=A$.
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9For example, the $1\times1$ matrix $A=-1$. – Andrés E. Caicedo Jan 20 '13 at 20:48
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I think you mean that the eigenvalues are all non-negative. If $A$ is symmetric and positive semi-definite, then this is a well-known exercise. Maybe that's what you meant? – cats Jan 20 '13 at 21:01