Page 133 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
116
which is relatively compact and has at most one cluster point (namely Q),
∗
converges to Q. Finally, H k = M k Q converges to MQ = H.
∗
k
Remark: There is as well a polar decomposition M = QH with the same
properties. We shall use one or the other depending on the context. We
warn the reader, however, that for a given matrix, the two decompositions
∗
do not coincide. For example, in M = HQ, H is the square root of MM ,
∗
though in M = QH, it is the square root of M M.
7.2 Exponential of a Matrix
The ground field is here k = CC. By restriction, we can also treat the case
k = IR.
For A in M n(CC), the series
∞
1 k
A
k!
k=0
converges normally (which means that the series of norms is convergent),
since for any matrix norm, we have
∞ #
# 1 # ∞
# 1 k
# A k # ≤ A =exp A .
k! k!
# #
k=0 k=0
Since M n (CC) is complete, the series is convergent, and the estimation above
shows that it converges uniformly on every compact set. Its sum, denoted
by exp A, thus defines a continuous map exp : M n (CC) → M n(CC), called
the exponential.When A ∈ M n (IR), we have exp A ∈ M n (IR).
Given two matrices A and B in general position, the binomial formula
k
is not valid: (A + B) does not necessarily coincide with
j=k
k j k−j
A B .
j
j=0
It thus follows that exp(A + B) differs in general from exp A · exp B.A
correct statement is the following.
Proposition 7.2.1 Let A, B ∈ M n (CC) be commuting matrices; that is,
AB = BA.Then exp(A + B)= (exp A)(exp B).
Proof
The proof proceeds in exactly the same way as for the exponential of
complex numbers. We observe that since the series defining the expo-
nential of a matrix is normally convergent, we may compute the product