Problem (i) Let A be a square nonsingular matrix of order na with LU factorization A = PLAUA and B be a square nonsingular matrix of order ng with LU factorization B = PLBUB. Find the LU factorization for A& B. (ii) Let A be a positive definite matrix of order nA with Cholesky factor GA and B be a positive definite matrix of order ng with Cholesky factor GB. Find the Cholesky factorization for A® B. (iii) Let A be an ma x na matrix with linear independent columns and QR factorization A = QẠRA, where Qa is an mĄ x na matrix with orthonormal columns and RĄ is an na x nĄ upper triangular matrix. B is similar defined with B = QBRB as its QR factorization. Find the QR factorization for A B. (iv) Let A be a square matrix of order nA with Schur decomposition A = UATAU, where UA is unitary and TA is upper triangular. Let B be a square matrix of order ng with Schur decomposition B = UBTBU, where UB is uni- tary and TB is upper triangular. Find the Schur decomposition for A8 B. (v) Let A be an MẠXNĄ matrix with singular value decomposition A = UĄ£AV and B be an mB × NB matrix with singular value decomposition B = U3EBV. Find the singular value decomposition for A O B. %3D %3D

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