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Say A is symmetric positive definite matrix , which means one necessary condition is diagonal entries of A are positive. If I do cholesky factorisation using outer product form, Can there be any positbility that $\sqrt{A[i][i]}$ will result in negative square roots ?

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There are no negative square roots; the root function of $c$ is defined to give you the positive value for $x$ that solves $x^2 = c$ if $c$ is a positive real number; in the complex plane, the square root is defined as the number with half the argument and square root the magnitude of c. Also, A is your original matrix, so $a_{i,i}$ is bound to be positive, as you've already stated; it's likely you'll have to rephrase your question – Marcus Müller Jan 11 '15 at 17:39

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