15.2. AUTOMATION WITH MATLAB 421

Example 15.2.7 Consider the symmetric matrix

A =

 1 2 32 2 13 1 4

 .

Let x= (1,1,1)T . How close is the Rayleigh quotient to some eigenvalue of A? Find theeigenvector and eigenvalue to several decimal places.

Everything is real and so there is no need to worry about taking conjugates. Therefore,the Rayleigh quotient is

(1 1 1

) 1 2 32 2 13 1 4

 1

11

3

=193

According to the above theorem, there is some eigenvalue of this matrix λ q such that

∣∣∣∣λ q−193

∣∣∣∣ ≤∣∣∣∣∣∣∣ 1 2 3

2 2 13 1 4

 1

11

− 193

 111

∣∣∣∣∣∣∣

√3

=1√3

 −13− 4

353

=

√19 +( 4

3

)2+( 5

3

)2

√3

= 1.2472

Could you find this eigenvalue and associated eigenvector? Of course you could. Thisis what the shifted inverse power method is all about.

Solve  1 2 3

2 2 13 1 4

− 193

 1 0 00 1 00 0 1

 x

yz

=

 111

In other words solve  −

163 2 3

2 − 133 1

3 1 − 73

 x

yz

=

 111

and divide by the entry which is largest, 3.8707, to get

u2 =

 .69925.49389

1.0

