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Matrix Computations

David Gleich

Purdue University

Fall 2017

Course number CS-51500

Tuesday and Thursday, 10:30-11:45am

Location Forney B124


Homework 8

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

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.

  1. Read this theorem. Explain it in your own words.

  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 matrix has a decomposition where the columns of are orthonormal eigenvectors with eigenvalues , and . Prove that the eigenvalues of a real symmetric matrix are real (you may not assume that the eigenvectors are real valued).

  2. Let be a real matrix with blocks and . Show that if is an eigenvector of with eigenvalue (where and are , then is an eigenvector with eigenvalue .

  3. Continuing part 2, assume that is an eigenvector of with eigenvalue . Show that (if nonzero) is an eigenvector of , and (if nonzero) is and eigenvector of , both with eigenvalue .

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