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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2025 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Spring 2025 • Question 2

(QR and Cholesky Factorizations). Let \(A\) be a nonsingular square matrix and let \(A = QR\) and \(A^*A = U^*U\) be the QR factorization of \(A\) and the Cholesky factorization of \(A^*A\), respectively. Assume that the usual normalizations \(r_{jj}, u_{jj} > 0\) are in effect. Is it true or false that \(R = U\)? Justify your answer with mathematical details.

Raw LaTeX

(QR and Cholesky Factorizations). Let \(A\) be a nonsingular square matrix
and let \(A = QR\) and \(A^*A = U^*U\) be the QR factorization of \(A\) and the Cholesky factorization
of \(A^*A\), respectively. Assume that the usual normalizations \(r_{jj}, u_{jj} > 0\) are in effect.
Is it true or false that \(R = U\)? Justify your answer with mathematical details.

Topic Tags

Cholesky Factorization and SPD Matrices Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra QR Factorization