272 CHAPTER 10. MARKOV PROCESSES
17. Suppose B =
(α c∗
b A
)is an (n+ 1)× (n+ 1) Hermitian nonnegative matrix where
α is a scalar and A is n × n. Show that α must be real, c = b, and A = A∗, A isnonnegative, and that if α = 0, then b = 0. Otherwise, α > 0.
18. ↑If A is an n× n complex Hermitian and nonnegative matrix, show that there existsan upper triangular matrix B such that B∗B = A. Hint: Prove this by induction. Itis obviously true if n = 1. Now if you have an (n+ 1)× (n+ 1) Hermitian nonnegative
matrix, then from the above problem, it is of the form
(α2 αb∗
αb A
), α real.
19. ↑ Suppose A is a nonnegative Hermitian matrix (all eigenvalues are nonnegative) whichis partitioned as
A =
(A11 A12
A21 A22
)where A11, A22 are square matrices. Show that det (A) ≤ det (A11) det (A22). Hint:Use the above problem to factor A getting
A =
(B∗
11 0∗
B∗12 B∗
22
)(B11 B12
0 B22
)
Next argue that A11 = B∗11B11, A22 = B∗
12B12 +B∗22B22. Use the Cauchy Binet theo-
rem to argue that det (A22) = det (B∗12B12 +B∗
22B22) ≥ det (B∗22B22) . Then explain
why
det (A) = det (B∗11) det (B
∗22) det (B11) det (B22)
= det (B∗11B11) det (B
∗22B22)
20. ↑ Prove the inequality of Hadamard. If A is a Hermitian matrix which is nonnegative(all eigenvalues are nonnegative), then det (A) ≤
∏iAii.