John Stillwell - Naive Lie Theory.pdf - Index of

4.5 The exponential of a square matrix 85
Now, summing the squares of both sides, we get
|AB| 2 = ∑|(i, j)-entry of AB| 2
i, j
≤ ∑
i, j
= ∑
i
(
|ai1 | 2 + ···+ |a in | 2)( |b 1 j | 2 + ···+ |b nj | 2)
(
|ai1 | 2 + ···+ |a in | 2) (
∑ |b1 j | 2 + ···+ |b nj | 2)
j
= |A| 2 |B| 2 , as required □
It follows from the submultiplicative property that |A m |≤|A| m . Along
with the triangle inequality |A +B|≤|A|+|B|, the submultiplicative property
enables us to test convergence of matrix infinite series by comparing
them with series of real numbers. In particular, we have:
Convergence of the exponential series. If A is any n × n real matrix, then
1 + A 1! + A2
2! + A3
+ ···, where 1 = n × n identity matrix,
3!
is convergent in R n2 .
Proof. It suffices to prove that this series is absolutely convergent, that is,
to prove the convergence of
|1| + |A|
1! + |A2 |
2! + |A3 |
3! + ···.
This is a series of positive real numbers, whose terms (except for the first)
are less than or equal to the corresponding terms of
1 + |A|
1! + |A|2
2! + |A|3
3! + ···
by the submultiplicative property. The latter series is the series for the real
exponential function e |A| ; hence the original series is convergent. □
Thus it is meaningful to make the following definition, valid for real,
complex, or quaternion matrices.