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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2015 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Fall 2015 • Question 3

Singular Value Decomposition (SVD)

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

b) Consider \(A\in \mathbb {C}^{m\times m}\) and show that \[ \lvert \det (A)\rvert =\prod _{i=1}^m\sigma _i, \] where \(\{\sigma _i\}\) are the singular values of \(A\).

Raw LaTeX

{Singular Value Decomposition (SVD)}

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

\textbf{b)} Consider \(A\in\C^{m\times m}\) and show that
\[
  \lvert\det(A)\rvert=\prod_{i=1}^m\sigma_i,
\]
where \(\{\sigma_i\}\) are the singular values of \(A\).

Topic Tags

Linear Programming and Optimization Singular Value Decomposition