Thread: symmetric matrices ,,

Let Sym(n) be the vector space of symmetric nⅹn matrices, and LT(n) the vector space of lower-triangular nⅹn 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 ??

Originally Posted by jin_nzzang

Let Sym(n) be the vector space of symmetric nⅹn matrices, and LT(n) the vector space of lower-triangular nⅹn 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 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.