Download MA1506 TUTORIAL 10

MA1506 TUTORIAL 10
Question 1
Wherever possible, diagonalize the matrices in questions 4/5 of Tutorial 9. [That is, write
them in the form PDP−1 , after finding P and D, where D is diagonal.] Using this, work
out their 4th powers.
[Answers: The first one cannot be diagonalized; the fourth powers of the others are
8 8
8 8
!
,
41 80
20 41
!
,
−7 24
−24 −7
!
0 0
0 1
,
!
.]
Question 2
The sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .......
in which each term is the sum of the two preceding ones, is called the Fibonacci sequence
[http://en.wikipedia.org/wiki/Fibonacci number ]. These numbers are very popular with
mathematicians, also with believers in the Da Vinci Code, and other nutcases. Let xk
denote the kth Fibonacci number, where x0 is 0. Show that
xk+1
xk+2
!
=
0 1
1 1
!
!
xk
xk+1
!
.
!
0 1
xk
, so Vk+1 = AVk , where A =
. By diagonalizing the
Define Vk =
xk+1
1 1
matrix, express Vk in terms of V0 . Hence prove Binet’s formula,
λk+ − λk−
xk = √
,
5
where
√
1
λ± = [1 ± 5].
2
Notice the mildly interesting fact that the two numbers λ± are irrational, and yet xk is a
whole number. Find limk→∞ xxk+1
. This limiting ratio is called the golden ratio, see [but
k
read with a skeptical mind] http://en.wikipedia.org/wiki/Golden ratio.
Question 3
The great white shark [http://en.wikipedia.org/wiki/Great white shark] likes to eat seals.
Let Σk denote the number of sharks in a given area at year k, and let σk denote the number
of seals in that area. In year k+1, the number of sharks depends on the number in year
k and also on the number of seals [the more seals there are in year k, the more food for
the sharks]. It is found that
Σk
σk
Σk+1 =
+
.
2
100
Notice that this means that the sharks would die out if there were no seals for them to
eat, which makes sense, since seals are their main food. Similarly the number of seals in
year k+1 is determined by the number of sharks and by the number of seals in year k. It
is found that
5σk 50Σk
−
.
σk+1 =
4
4
Notice the minus sign: seals tend to disappear into the mouths of sharks. Notice too
that the seal population would grow uncontrollably if the sharks were not kind enough to
remove surplus seals. Suppose we start with 50 sharks and 1600 seals in year zero. How
many sharks and seals will we have in the long run? [Answer: 14 sharks and 700 seals.]
What is the answer if the 50/4 in the second equation is changed to 51/4? Comment.
Question 4
Let B be a diagonalizable matrix. Prove that det(eB ) = eT r(B) . Verify this for the 2 × 2
rotation matrix in question 7 of Tutorial 8.
Question 5
Romeo is fed up with Juliet’s fickle ways, and decides to dump her. He runs off to Venice
in disgust and takes up instead with Desdemona, a more straightforward kind of girl
whose love for Romeo grows when he likes her, and decreases when he doesn’t. Romeo
remains as before. Consider the special case in which Desdemona has no feelings either
way for Romeo when the latter first falls in love with her. Will Desdemona’s love for
Romeo ever exceed his love for her? Can the two star-crossed lovers ever have a stable
relationship?
Question 6
Meanwhile Juliet, in a rage, leaves Verona [her home town, where she met Romeo] and
follows Romeo to Venice, where she discovers Romeo madly in love with Desdemona.
Heartbroken, she immediately falls in love with Othello, who is a fickle lover like Juliet:
that is, if a girl likes him, his opinion of her immediately drops, but his opinion begins to improve the moment she shows signs of hating him1 . Predict the outcome, assuming that Othello initially likes Juliet too. Draw some typical phase plane diagrams.
http://en.wikipedia.org/wiki/Othello
Question 7
One of the many virtues of Laplace transforms is that they turn calculus problems into
algebra problems. That works also for systems of simultaneous first-order ordinary differential equations, as follows. Suppose you want to solve the following pair of simultaneous
equations:
dy
dx
= 4x − 5y,
= 2x − 2y
dt
dt
with x(0) = 0 and y(0) = 2. You can write this as
*
dv
*
= Bv,
dt
1
He loves not wisely, and maybe not too well either.
2
*
where v(t) is a column vector built from x(t) and y(t), and B is the matrix of coefficients.
Now the Laplace transform of a vector function of time is defined in the obvious way
*
*
[treating each component separately] so if we let V (s) be the Laplace transform of v(t),
then we have
*
*
*
sV (s) − v(0) = B V (s).
*
This is now an algebra problem: by inverting a matrix you can find V (s), and then taking
*
the inverse Laplace transform you can find v(t). Do this and find x(t), y(t). [Answer: x(t)
= −10sin(t)et , y(t) = 2et cos(t) − 6et sin(t). Compare with the example on pages 14-16 of
Chapter 7.]
3