370 CHAPTER 15. NUMERICAL METHODS, EIGENVALUES

14. Use the QR algorithm to approximate the eigenvalues of the symmetric matrix 1 2 32 −8 13 1 0



15. Try to find the eigenvalues of the matrix

 3 3 1−2 −2 −10 1 0

 using the QR algo-

rithm. It has eigenvalues 1, i,−i. You will see the algorithm won’t work well. ▶

16. Let q(λ ) = a0 +a1λ + · · ·+an−1λn−1 +λ

n. Now consider the companion matrix,

C ≡

0 · · · 0 −a0

1 0 −a1. . . . . .

...0 1 −an−1

Show that q(λ ) is the characteristic equation for C. Thus the roots of q(λ ) are theeigenvalues of C. You can prove something similar for

C =

−an−1 −an−2 · · · −a0

1. . .

1

Hint: The characteristic equation is

det

λ · · · 0 a0

−1 λ a1. . . . . .

...0 −1 λ +an−1

Expand along the first column. Thus

λ det

λ · · · 0 a1

−1 λ a2. . . . . .

...0 −1 λ +an−1

+det

0 0 · · · a0

−1 λ · · · a2...

. . ....

0 −1 λ +a3

Now use induction on the first term and for the second, note that you can expandalong the top row to get

(−1)n−2 a0 (−1)n = a0.