362 CHAPTER 14. NUMERICAL METHODS, EIGENVALUES

Simplifying the matrix on the left, I must solve 12 −14 11

−4 11 −4

3 6 4

 x

y

z

 =

 1

1

1

and then divide by the entry which has largest absolute value to obtain

u2 =

 1.0

. 184

−. 76

Now solve  12 −14 11

−4 11 −4

3 6 4

 x

y

z

 =

 1.0

. 184

−. 76

and divide by the largest entry, 1. 051 5 to get

u3 =

 1.0

.0 266

−. 970 61

Solve  12 −14 11

−4 11 −4

3 6 4

 x

y

z

 =

 1.0

.0 266

−. 970 61

and divide by the largest entry, 1. 01 to get

u4 =

 1.0

3. 845 4× 10−3

−. 996 04

 .

These scaling factors are pretty close after these few iterations. Therefore, the predictedeigenvalue is obtained by solving the following for λ.

1

λ+ 7= 1.01

which gives λ = −6. 01. You see this is pretty close. In this case the eigenvalue closest to−7 was −6.

How would you know what to start with for an initial guess? You might apply Ger-schgorin’s theorem. However, sometimes you can begin with a better estimate.

Example 14.1.5 Consider the symmetric matrix A =

 1 2 3

2 1 4

3 4 2

 . Find the middle

eigenvalue and an eigenvector which goes with it.