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)
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.
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.
Read this theorem. Explain it in your own words.
Write a function to plot the Gershgorin circles in Julia.
Plot the Gershgorin circles for the matrix:
n = 10
on = ones(Int64,n)
A = spdiagm((-2*on,4*on,-2*on),(-1,0,1))
Use Gershgorin circles to prove strictly diagonally dominant matrices are invertible.
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.)
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).
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 .
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 .
Show that the Jacobi iteration converges for 2-by-2 symmetric, positive, definite systems.