Page 239 - Applied Probability

10. Molecular Phylogeny
10. Let A and B be the 2 × 2 real matrices

A =
λ
b
1
a
Show that
a −b , B = λ 0 . 225
cos b − sin b B λ 10
A
a
e = e , e = e .
sin b cos b 11

(Hints: Note that 2 × 2 matrices of the form a −b are isomorphic
b a

to the complex numbers under the correspondence a −b ↔ a + bi.
b a
For the second case write B = λI + C.)
11. Define matrices

a 0 b 1
A = , B = .
1 a 0 b
Show that AB = BA and that

11
a+b
A B
e e = e
12

10 01
e A+B = e a+b cosh(1) + sinh(1) .
01 10
A B
A
B
Hence, e e = e A+B . (Hint: Use Problem 10 to calculate e and e .
2
For e A+B write A + B =(a + b)I + R with R satisfying R = I.)
A
12. Prove that det(e )= e tr(A) , where tr is the trace function. (Hint:
Since the diagonalizable matrices are dense in the set of matrices
[11], by continuity you may assume that A is diagonalizable.)
13. For the nucleotide substitution model of Section 10.5, prove formally
that P(t) has the same pattern for equality of entries as Λ. For exam-
k
ple, p AC (t)= p GC (t). (Hint: Prove by induction that Λ has the same
pattern as Λ. Then note the matrix exponential definition (10.7).)
14. For the nucleotide substitution model of Section 10.5, show that Λ
has eigenvalues 0, −(γ+λ+δ+κ), −(α+ +γ+λ), and −(δ+κ+β+σ)
and corresponding right eigenvectors
t

1 = 1, 1, 1, 1
t

c 5 c 5
u = 1, 1, − , −
c 2 c 2
α(c 5 − c 3 )+ κc 2 − (c 5 − c 3 ) − δc 2 t
v = , , 1, 1
δ(c 3 − c 2 ) − c 5 δ(c 3 − c 2 ) − c 5
t

β(c 2 − c 6 )+ λc 5 −σ(c 2 − c 6 ) − γc 5
w = 1, 1, , ,
γ(c 6 − c 5 ) − σc 2 γ(c 6 − c 5 ) − σc 2