Linear Algebra, Theory And Applications, 2012a

194 SPECTRAL THEORY
Now you finish the argument. To show uniqueness in the second part, suppose
y ′ =(a + ib) y, y (t 0 )=0
and verify this requires y (t) = 0. To do this, note
and that |y| 2 (t 0 )=0and
y ′ =(a − ib) y, y (t 0 )=0
d
dt |y (t)|2 = y ′ (t) y (t)+y ′ (t) y (t)
=(a + ib) y (t) y (t)+(a − ib) y (t) y (t) =2a |y (t)| 2 .
Thus from the first part |y (t)| 2 =0e −2at =0. Finally observe by a simple computation
that (7.20) is solved by (7.21). For the last part, write the equation as
y ′ − ay = f
and multiply both sides by e −at andthenintegratefromt 0 to t using the initial
condition.
38. Now consider A an n × n matrix. By Schur’s theorem there exists unitary Q such that
Q −1 AQ = T
where T is upper triangular. Now consider the first order initial value problem
x ′ = Ax, x (t 0 )=x 0 .
Show there exists a unique solution to this first order system. Hint: Let y = Q −1 x
and so the system becomes
y ′ = T y, y (t 0 )=Q −1 x 0 (7.22)
Now letting y =(y 1 , ··· ,y n ) T , the bottom equation becomes
y ′ n = t nn y n ,y n (t 0 )= ( Q −1 x 0
)n .
Then use the solution you get in this to get the solution to the initial value problem
which occurs one level up, namely
y ′ n−1 = t (n−1)(n−1) y n−1 + t (n−1)n y n ,y n−1 (t 0 )= ( Q −1 x 0
)
n−1
Continue doing this to obtain a unique solution to (7.22).
39. Now suppose Φ (t) isann × n matrix of the form
where
Φ(t) = ( x 1 (t) ··· x n (t) ) (7.23)
x ′ k (t) =Ax k (t) .
Explain why
Φ ′ (t) =AΦ(t)
ifandonlyifΦ(t) is given in the form of (7.23). Also explain why if c ∈ F n , y (t) ≡
Φ(t) c solves the equation y ′ (t) =Ay (t) .