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Understanding LU and Cholesky Decompositions for Matrix Operations

Learn about LU and Cholesky Decompositions in matrix theory. Explore their applications, elementary row operations, and implications for nonsingular and symmetric matrices.

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Understanding LU and Cholesky Decompositions for Matrix Operations

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  1. LU Decomposition& Cholesky Decomposition 2018/10/30 You-Jia Sun

  2. LU Decomposition A is a n-by-n matrix A=LU L is lower triangular matrix; U is upper triangular matrix. A L U

  3. Elementary row operation • Row switching: Rowi<-> Rowj • Row multiplication: kRowi-> Rowi, where k ≠ 0 • Row substitution: Replace: Rowi - lRowj-> Rowi, where i ≠ j • The row calculation result, called U mtarix, is an upper matrix. • If all of the diagonal elements of U matrix is not zero, A is nonsingular matrix. U

  4. Elementary matrix Eij(i,j) = -lij • L is an unitriangular matrix and its elements below diagonal are -lij

  5. LDU decomposition A L U = * = * * D U L

  6. Cholesky Decomposition A = LLT = * = symmetric

  7. Change a vector of independent r.v. into dependent variables A random vector Z, satisfied cov(Z) = I We want to get vector X = (x1,…,xn)Tsatisfied cov(X) = ∑ By Cholesky Decomposition,∑ = LLT,X=LZ Cov(X) = cov(LZ) = Lcov(Z)LT=LLT= ∑

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