286 CHAPTER 12. SPECTRAL THEORY
Explain why (An +an−1An−1 + · · ·+a1A+a0I
)v = 0
If A is nondefective, give a very easy proof of the Cayley Hamilton theorem basedon this. Recall this theorem says A satisfies its characteristic equation,
An +an−1An−1 + · · ·+a1A+a0I = 0.
50. Suppose an n×n nondefective matrix A has only 1 and −1 as eigenvalues. Find A12.
51. Suppose the characteristic polynomial of an n×n matrix A is 1−λn. Find Amn where
m is an integer. Hint: Note first that A is nondefective. Why?
52. Sometimes sequences come in terms of a recursion formula. An example is theFibonacci sequence.
x0 = 1 = x1, xn+1 = xn + xn−1
Show this can be considered as a discreet dynamical system as follows.(xn+1
xn
)=
(1 11 0
)(xn
xn−1
),
(x1
x0
)=
(11
)Now use the technique of Problem 48 to find a formula for xn. This was done in thechapter. Next change the initial conditions to x0 = 0,x1 = 1 and find the solution.
53. Let A be an n×n matrix having characteristic polynomial
det(λ I−A) = λn +an−1λ
n−1 + · · ·+a1λ +a0
Show that a0 = (−1)n det(A).
54. Find
(32 1− 1
2 0
)35
. Next find
limn→∞
(32 1− 1
2 0
)n
55. Find eA where A is the matrix
(32 1− 1
2 0
)in the above problem.
56. Consider the dynamical system
(x(n+1)y(n+1)
)=
(.8 .8−.8 .8
)(x(n)y(n)
). Show
eigenvalues and eigenvectors are 0.8+ 0.8i←→
(−i1
),0.8− 0.8i←→
(i1
).
Find a formula for the solution to the dynamical system for given initial condition(x0,y0)
T . Show that the magnitude of (x(n) ,y(n))T must diverge provided the initialcondition is not zero. Next graph the vector field for(
.8 .8−.8 .8
)(xy
)−
(xy
)