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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2022 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Fall 2022 • Question 2

(Properties via SVD).

  1. Consider \(A \in \mathbb {C}^{m \times n}\). Define what we mean by the singular value decomposition of \(A\).
  2. Prove that any matrix in \(\mathbb {C}^{m \times n}\) is the limit of a sequence of matrices of full rank. Use the 2-norm for your proof.

Raw LaTeX

(Properties via SVD).
\begin{enumerate}

\item
Consider \(A \in \mathbb{C}^{m \times n}\). Define what we mean by the singular value decomposition of \(A\).


\item
Prove that any matrix in \(\mathbb{C}^{m \times n}\) is the limit of a sequence of matrices of full rank. Use
the 2-norm for your proof.

\end{enumerate}

Topic Tags

Linear Programming and Optimization Matrix Norms and Induced Norms Matrix Rank and Fundamental Linear Algebra Singular Value Decomposition