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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2013 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2013 • Question 1
**Application of Matrix Factorizations** **a)** Show that for any real \(n\times n\) matrix \(A\) and any \(\epsilon>0\), there is a nonsingular matrix \(B\) for which \(\|A-B\|_2<\epsilon\). **b)** Show that for any real \(n\times n\) matrix \(A\) and any \(\epsilon>0\), there is a diagonalizable matrix \(B\) for which \(\|A-B\|_2<\epsilon\).

Raw LaTeX

{Application of Matrix Factorizations}

\partitem{a} Show that for any real \(n\times n\) matrix \(A\) and any
\(\epsilon>0\), there is a nonsingular matrix \(B\) for which
\(\|A-B\|_2<\epsilon\).

\partitem{b} Show that for any real \(n\times n\) matrix \(A\) and any
\(\epsilon>0\), there is a diagonalizable matrix \(B\) for which
\(\|A-B\|_2<\epsilon\).

Topic Tags

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