Page 75 - Matrices theory and applications

3. Matrices with Real or Complex Entries
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account of this question, one may read the first section of Fulton’s
and Bhatia’s articles [16, 6]. For another partial result, see Exercise
19 of Chapter 5 (theorem of Lidskii).
12. Let A be a Hermitian matrix of size n × n whose eigenvalues are
α 1 ≤ ··· ≤ α n .Let B be a Hermitian positive semidefinite matrix.
Let γ 1 ≤ ··· ≤ γ n be the eigenvalues of A + B. Show that γ k ≥ α k .
13. Let M, N be two Hermitian matrices such that N and M − N are
positive semidefinite. Show that det N ≤ det M.
14. Let A ∈ M p (CC), C ∈ M q (CC)begiven with p, q ≥ 1. Assume that

A B
M :=
B ∗ C
is Hermitian positive definite. Show that det M ≤ (det A)(det C). Use
the previous exercise and Proposition 8.1.2.
15. For M ∈ HPD n ,wedenote by P k (M) the product of all the principal
minors of order k of M.There are

n
k
such minors.
Applying Proposition 3.4.2 to the matrix M −1 , show that
P n (M) n−1 ≤ P n−1 (M),

andtheningeneral that
k
P k+1 (M) ≤ P k (M) n−k .
16. Let d : M n (IR) → IR + be a multiplicative function; that is,

d(MN)= d(M)d(N)
for every M, N ∈ M n (IR). If α ∈ IR, define δ(α):= d(αI n ) 1/n .
Assume that d is not constant.
(a) Show that d(0 n )=0 and d(I n ) = 1. Deduce that P ∈ GL n (IR)
implies d(P) =0 and d(P −1 )= 1/d(P). Show, finally, that if M
and N are similar, then d(M)= d(N).
(b) Let D ∈ M n (IR) be diagonal. Find matrices D 1 ,... ,D n−1 ,sim-
ilar to D,suchthat DD 1 ··· D n−1 =(det D)I n . Deduce that
d(D)= δ(det D).
(c) Let M ∈ M n (IR) be a diagonalizable matrix. Show that d(M)=
δ(det M).
(d) Using the fact that M T is similar to M, show that d(M)=
δ(det M) for every M ∈ M n (IR).