Suppose that a is a symmetric positive definite. Show that the matrix â’a is not symmetric positive definite but that the matrix at is symmetric positive definite.
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Suppose A is symmetric and real. Then is it diagonalisable by an orthogonal matrix. That is A=UΛUT, with U orthogonal and Λ diagonal. Since A is positive semidefinite, all entries of Λ are non negative hence have a square root. Denote the diagonal matrix of square roots by Λ−−√. Let B=UΛ−−√UT, then it is easy to check that B2=A.
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A matrix is positive definite if it's symmetric and all its pivots are positive. where Ak is the upper left k x k submatrix. All the pivots will be pos itive if and only if det(Ak) > 0 for all 1 k n. So, if all upper left k x k determinants of a symmetric matrix are positive, the matrix is positive definite.
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