Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2016 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2016 • Question 1

**Singular Value Decomposition (SVD):** **a)** Prove the following statement: Singular Value Decomposition: Any matrix \(A\in\mathbb{C}^{m\times n}\) can be factored as \(A=U\Sigma V^*\), where \(U\in\mathbb{C}^{m\times m}\) and \(V\in\mathbb{C}^{n\times n}\) are unitary and \(\Sigma\in\mathbb{R}^{m\times n}\) is a rectangular matrix whose only nonzero entries are non-negative entries on its diagonal. **b)** Use the SVD to prove that any matrix in \(\mathbb{C}^{n\times n}\) is the limit of a sequence of matrices of full rank.

{Singular Value Decomposition (SVD):}

\partitem{a} Prove the following statement:

\begin{quote}
Singular Value Decomposition: Any matrix \(A\in\C^{m\times n}\) can be factored
as \(A=U\Sigma V^*\), where \(U\in\C^{m\times m}\) and \(V\in\C^{n\times n}\) are
unitary and \(\Sigma\in\R^{m\times n}\) is a rectangular matrix whose only
nonzero entries are non-negative entries on its diagonal.
\end{quote}

\partitem{b} Use the SVD to prove that any matrix in \(\C^{n\times n}\) is the
limit of a sequence of matrices of full rank.

Topic Tags

Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra Orthogonal and Unitary Matrices Singular Value Decomposition