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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2024 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Fall 2024 • Question 3

(Eigenvalues and Eigenvectors).

  1. State the Rayleigh Quotient Iteration Algorithm to find eigenvalues/eigenvectors for general real symmetric matrix \(A \in \mathbb {R}^{m \times m}\).
  2. Prove that the spectrum (the set of all eigenvalues) of any matrix \(A \in \mathbb {C}^{m \times m}\) is contained in the union of the following \(m\) disks, \(D_i\), in the complex plane: \[ D_i = \left \{z \in \mathbb {C} : |z - a_{ii}| \le \sum _{\substack {j=1\\ j\ne i}}^m |a_{ij}|\right \},\qquad 1 \le i \le m. \]

Raw LaTeX

(Eigenvalues and Eigenvectors).
\begin{enumerate}

\item
State the Rayleigh Quotient Iteration Algorithm to find eigenvalues/eigenvectors for
general real symmetric matrix \(A \in \mathbb{R}^{m \times m}\).


\item
Prove that the spectrum (the set of all eigenvalues) of any matrix \(A \in \mathbb{C}^{m \times m}\) is contained
in the union of the following \(m\) disks, \(D_i\), in the complex plane:
\[
D_i = \left\{z \in \mathbb{C} : |z - a_{ii}| \le
\sum_{\substack{j=1\\ j\ne i}}^m |a_{ij}|\right\},\qquad 1 \le i \le m.
\]


\end{enumerate}

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Fixed-Point Iteration Linear Programming and Optimization Rayleigh Quotient Iteration