1. Numerical Linear Algebra

Eigenvalue Computation — Quiz

Test your understanding of eigenvalue computation with 5 practice questions.

Practice Questions

Question 1

Which spectral transformation underlies the shift-and-invert method for computing eigenvalues of a matrix $A$ closest to a specified shift $\mu$?

Question 2

Given the matrix $A=\begin{pmatrix}4 & 1\\1 & 3\end{pmatrix},$ shift $\mu=2,$ and initial vector $x_0=\begin{pmatrix}1\\1\end{pmatrix},$ perform one shift-and-invert iteration: compute $y=(A-\mu I)^{-1}x_0$ and normalize to obtain $x_1$. What is $x_1$?

Question 3

After obtaining a dominant eigenpair $(\lambda_1,v_1)$ of a symmetric matrix $A$, the deflated matrix $A'$ used to find subsequent eigenvalues is given by which formula?

Question 4

What is the approximate floating-point operation count to reduce a dense symmetric $n\times n$ matrix to tridiagonal form using Householder reflections?

Question 5

In the implicitly shifted QR algorithm for symmetric tridiagonal matrices, what is the purpose of the bulge-chasing procedure?