Page 147 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
130
−1
∗
Let now define V R := M U R S
.From above, wehave
−1
−1
∗
V V R = S
∗
= I r .
U MM U R S
∗
R
R
This means that the columns v 1 ,... , v r of V R constitute an orthonormal
family.
Noting that these column vectors belong to R(M ), that is, to (ker M) ,
∗
⊥
a subspace of codimension r,we see that {v 1 ,... , v r } can be extended to
m
an orthonormal basis {v 1 ,... , v m } of CC ,where v r+1 ,... belong to ker M.
Let V =: (V R ,V K ) be the unitary matrix whose columns are v 1 ,...
∗
We now compute blockwise the product U MV .From MV K =0and
∗
M U ∗ =0, we get
K

∗
U MV R 0
R
U MV = .
∗
0 0
Finally, we obtain
∗
∗
∗
∗
U MV R = U MM U R S −1 = U U R S = S.
R
R
R
7.8 Exercises
1. Show that the square root map from HPD n into itself is continuous.
2. Let M ∈ M n(k) be given, with k = IR or CC. Show that there ex-
ists a polynomial P ∈ k(X), of degree at most n − 1, such that
P(M)= exp M. However, show that this polynomial cannot be
chosen independently of the matrix.
Compute this polynomial when M is nilpotent.
3. For t ∈ IR, define Pascal’s matrix P(t)by p ij (t)= 0if i< j (the
matrix is lower triangular) and

i − 1
i−j
p ij (t)= t
j − 1
otherwise. Let us emphasize that for just this once in this book, P
is an infinite matrix, meaning that its indices range over the infinite
∗
set IN . Compute P (t) and deduce that there exists a matrix L such

that P(t)= exp(tL). Compute L explicitly.
1
4. Let I be an interval of IR and t → P(t)beamapofclass C with
2
values in M n (IR) such that for each t, P(t) is a projector: P(t) =
P(t).
(a) Show that the rank of P(t) is constant.
(b) Show that P(t)P (t)P(t)=0 n .