Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2018 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2018 • Question 1
**Singular values and eigenvalues** Let \(A\in\mathbb{C}^{N\times N}\). Prove the following statements: **(a)** The set of squared singular values of \(A\) equals the set of eigenvalues of \(A^*A\). **(b)** Assume that \(A\) is diagonalizable and that every eigenvalue \(\lambda\) of \(A\) satisfies \(|\lambda|<1\). Then: \[ \lim_{n\to\infty}\|A^n\|=0. \]

Raw LaTeX

{Singular values and eigenvalues}

Let \(A\in\C^{N\times N}\). Prove the following statements:

\partitem{a} The set of squared singular values of \(A\) equals the set of
eigenvalues of \(A^*A\).

\partitem{b} Assume that \(A\) is diagonalizable and that every eigenvalue
\(\lambda\) of \(A\) satisfies \(|\lambda|<1\). Then:
\[
  \lim_{n\to\infty}\|A^n\|=0.
\]

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra Singular Value Decomposition