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

302 Chapter 8 Matrix algebra
(b) The characteristic equation issolved toyield the eigenvalues:
λ 3 +2λ 2 −λ−2=0
Factorizing yields
(λ+2)(λ+1)(λ−1)=0
from which λ = −2,−1,1.
The eigenvalues are λ = −2,−1,1.
EXERCISES8.11.2
1 Calculate (i) the characteristic equation (ii) the
eigenvalues ofthe systemAX = λX,
whereAisgiven by
( ) ( )
5 6 −3 4
(a) (b)
(c)
2 1
( 7 −2
1 4
)
(d)
−4 5
( ) 1 3
4 −1
2 Calculate (i) the characteristic equation (ii) the
eigenvalues ofthe following 3 ×3 matrices:
⎛ ⎞
1 −12
(a) ⎝−3 −23⎠
2 −11
(b)
(c)
(d)
(e)
⎛ ⎞
10 −1
⎝31 4⎠
02 2
⎛ ⎞
21 2
⎝−11 −1⎠
83 0
⎛ ⎞
−2 62
⎝ 0 34⎠
3 −35
⎛ ⎞
3 −21
⎝ 2 −43⎠
16 −41
Solutions
1 (a) (i) λ 2 −6λ−7=0
(ii) λ = −1,7
(b) (i) λ 2 −2λ+1=0
(ii) λ = 1 (twice)
(c) (i) λ 2 −11λ+30=0
(ii) λ=5,6
(d) (i) λ 2 −13=0
(ii) λ = − √ 13, √ 13
2 (a) (i) −λ 3 +7λ+6=0
(ii) λ = −2, −1,3
(b) (i) λ 3 −4λ 2 −3λ+12=0
(ii) λ = − √ 3, √ 3,4
(c) (i) λ 3 −3λ 2 −10λ+24=0
(ii) λ = −3,2,4
(d) (i) λ 3 −6λ 2 +5λ=0
(ii) λ=0,1,5
(e) (i) λ 3 −13λ+12=0
(ii) λ = −4,1,3
TechnicalComputingExercises8.11.2
Thecalculationofeigenvalues andeigenvectors is
usuallyperformedby abuilt-infunction. Forexample,
in MATLAB ® the functioneigcan used to calculate
the eigenvalues.
To produce asolution to question 1.(a)in the previous
exercise wewould type:
A=[56;21];
eig(A)