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$Old$ November 3rd, 2009, 11:52 PM

JohnLeee $JohnLeee is offline$

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$Default$ Symmetry and Eigenvalue

I don't understand how eigenvalues being non negative helps

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$Old$ November 4th, 2009, 02:31 AM

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I don't understand how eigenvalues being non negative helps

If A is symmetric then it is diagonalisable, $A = PDP^{-1}$ , where D is a diagonal matrix whose diagonal elements are the eigenvalues of A. If these are all non-negative then D has a square root E, namely the diagonal matrix whose diagonal elements are the square roots of those of D. Let $B = PEP^{-1}$ . Then $B^2=A$ .

Edit. I should have said that A is orthogonally diagonisable, so P can be taken to be an orthogonal matrix, $P^{-1} = P^{\textsc t}$ . Then $B = PEP^{\textsc t}$ , which ensures that B is symmetric.

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Last edited by Opalg; November 4th, 2009 at 08:56 AM.

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