15.4. EXERCISES 439

9. Using Gerschgorin’s theorem, find upper and lower bounds for the eigenvalues of

A =

 3 2 32 6 43 4 −3

 .

10. Tell how to find a matrix whose characteristic polynomial is a given monic polyno-mial. 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 upper Hes-senberg matrix is followed. Show the resulting upper Hessenberg matrix is actuallyequal to 0 on the top as well as the bottom.