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.