386 CHAPTER 14. NUMERICAL METHODS, EIGENVALUES

9. Using Gerschgorin’s theorem, find upper and lower bounds for the eigenvalues of A = 3 2 3

2 6 4

3 4 −3

 .

10. Tell how to find a matrix whose characteristic polynomial is a given monic polynomial.This is called a companion matrix. Find the roots of the polynomial x3+7x2+3x+7.

11. Find the roots to x4 + 3x3 + 4x2 + x+ 1. It has two complex roots.

12. Suppose A is a real symmetric matrix and the technique of reducing to an upperHessenberg matrix is followed. Show the resulting upper Hessenberg matrix is actuallyequal to 0 on the top as well as the bottom.