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Math Help - symmetric matrices ,,

  1. #1
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    symmetric matrices ,,

    Let Sym(n) be the vector space of symmetric nn matrices, and LT(n) the vector space of lower-triangular nn matrices. Define F : LT(n) Sym(n) by F(A) = ATA, where AT is the transpose of A. Show that there exists an open set U about the identity matrix in LT(n) and an open set V about the identity matrix in Sym(n), such that for each symmetric matrix B V there is a unique lower triangular matrix A U such that F(A) = B.


    could anyone help me to solve this problem ??
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    Quote Originally Posted by jin_nzzang View Post
    Let Sym(n) be the vector space of symmetric nn matrices, and LT(n) the vector space of lower-triangular nn matrices. Define F : LT(n) Sym(n) by F(A) = ATA, where AT is the transpose of A. Show that there exists an open set U about the identity matrix in LT(n) and an open set V about the identity matrix in Sym(n), such that for each symmetric matrix B V there is a unique lower triangular matrix A U such that F(A) = B.

    could anyone help me to solve this problem ??
    Let U be the set of matrices in LT(n) with strictly positive elements on the diagonal, and let V be the set of positive definite matrices in Sym(n). Then U and V are open sets containing the identity. The map F(A) = A^{\,\textsc{t}}A clearly takes U into V. Its inverse is the Cholesky decomposition, and there is a theorem (proved here) which states that this takes V to U.
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