
Homework 8

Please answer the following questions in complete sentences in a typed, clearly prepared manuscript and submit the solution by the due date on Blackboard (Monday, November 6th, 2017, around 4-5 am)

Problem 0: Homework checklist

• Please identify anyone, whether or not they are in the class, with whom you discussed your homework. This problem is worth 1 point, but on a multiplicative scale.

• Make sure you have included your source-code and prepared your solution according to the most recent Piazza note on homework submissions.

Problem 1: Gershgorin disks

Theorem 7.2.1 in Golub and van Loan states the Gershgorin circle theorem. This theorem provides an easy check to find regions containing the eigenvalues of a matrix.

2. Write a function to plot the Gershgorin circles in Julia.

3. Plot the Gershgorin circles for the matrix:

  n = 10
on = ones(Int64,n)
A = spdiagm((-2*on,4*on,-2*on),(-1,0,1))

4. Use Gershgorin circles to prove strictly diagonally dominant matrices are invertible.

Problem 2: More on Eigenvalues and convergence theory

Each of these has a painless, 2 to 4 sentence proof (if they cause lots of difficulty, try thinking about them in a different way.)

1. It was stated in class that a real symmetric $n \times n$ matrix $\mM$ has a decomposition $\mM = \mX \Lambda \mX^*$ where the columns of $\mX$ are orthonormal eigenvectors $\vx_i$ with eigenvalues $\lambda_i$, and $\Lambda = \text{diag}(\lambda_1, ..., \lambda_n)$. Prove that the eigenvalues $\lambda_i$ of a real symmetric matrix are real (you may not assume that the eigenvectors $\vx_i$ are real valued).

2. Let $\mG = \bmat{ 0 & \mB \\ \mA & 0}$ be a real $2n \times 2n$ matrix with $n \times n$ blocks $\mA$ and $\mB$. Show that if $\bmat{\vx \\ \vy}$ is an eigenvector of $\mG$ with eigenvalue $\lambda$ (where $\vx$ and $\vy$ are $n \times 1$, then $\bmat{\vx \\ -\vy}$ is an eigenvector with eigenvalue $-\lambda$.

3. Continuing part 2, assume that $\bmat{\vx \\ \vy}$ is an eigenvector of $\mG$ with eigenvalue $\lambda$. Show that $\vx$ (if nonzero) is an eigenvector of $\mB\mA$, and $\vy$ (if nonzero) is and eigenvector of $\mA \mB$, both with eigenvalue $\lambda^2$.

4. Show that the Jacobi iteration converges for 2-by-2 symmetric, positive, definite systems.