082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.11 Eigenvalues and eigenvectors 305
The concept ofeigenvectors iseasilyextended tomatrices ofhigher order.
Example8.48 Determine the eigenvectors of
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 2 0 x x
⎝−1−1 1⎠
⎝y⎠ = λ ⎝y⎠
3 2−2 z z
The eigenvalues werefound inExample 8.45.
Solution FromExample8.45theeigenvaluesare λ = −2,−1,1.Weconsidereacheigenvaluein
turn.
λ=−2
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 2 0 x x
⎝−1−1 1⎠
⎝y⎠ = −2⎝y⎠
3 2−2 z z
⎡⎛
⎞ ⎛ ⎞⎤⎛
⎞ ⎛ ⎞
1 2 0 100 x 0
⎣⎝−1−1 1⎠ +2⎝010⎠⎦
⎝y⎠ = ⎝0⎠
3 2−2 001 z 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
320 x 0
⎝−111⎠
⎝y⎠ = ⎝0⎠
320 z 0
We note that the first and lastrows areidentical. So wehave
3x +2y =0
−x+y +z=0
Solving these equations gives
x=t, y=− 3 2 t, z=5 2 t
Hence the corresponding eigenvector is
⎛ ⎞
1
− 3 X =t ⎜ 2⎟
⎝ ⎠
5
2
λ=−1
We have
⎡⎛
⎞
1 2 0
⎛ ⎞⎤⎛
⎞
100 x
⎣⎝−1−1 1⎠ + ⎝010⎠⎦
⎝y⎠ =
3 2−2 001 z
⎛ ⎞ ⎛ ⎞
22 0 x
⎝−10 1⎠
⎝y⎠ =
32−1 z
⎛ ⎞
0
⎝0⎠
0
⎛ ⎞
0
⎝0⎠
0