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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2012 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Spring 2012 • Question 4

Spectral Radius

Let \(A\in \mathbb {C}^{m\times m}\) and \(\|\cdot \|\) denote any operator norm on \(m\times m\) matrices. Show that \[ \lim _{k\to \infty }\|A^k\|^{1/k}=\rho (A). \]

Raw LaTeX

{Spectral Radius}

Let \(A\in\C^{m\times m}\) and \(\|\cdot\|\) denote any operator norm on
\(m\times m\) matrices. Show that
\[
  \lim_{k\to\infty}\|A^k\|^{1/k}=\rho(A).
\]

\vspace{1.2\baselineskip}

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Matrix Norms and Induced Norms