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Unformatted text preview: act and the Euclidean norm is continuous. Therefore
attains a maximum value on
the sphere and
is bounded.
A series
in a normed vector space is absolutely convergent if the series
This condition implies that
converges. (Hirsch and Smale, 1974) Recall the Taylor series for
about
from calculus:
the exponential of an operator in the same way: converges. . We formally define [7] Lemma 2.1. If T is a bounded linear operator then
Proof. By [6] is as well. and the series of real numbers converges to . Therefore the series [7] converges absolutely by the comparison test. It also follows that
.□
Properties of the exponential The first 6 of these are from Meiss. His Roman numerals are
given but in my order. I have added a property (vii).
[i] . This is a direct consequence of the definition of the exponential.
[v] If then 10
Notice
[vi] If and insert in the definition of the exponential.
is an eigenvector of with eigenvalue Notice and let then . . [iv] If B is nonsingular then . Using the definition of the exponential
.
[iii] If A and B commute then (the law of exponents!) See exercise #6. The idea of part (a) is that if and are real numbers then the series for their
exponentials can be manipulated to show that
. Since matrices and commute,
their series can be manipulated in the same way. However, the manipulation is complicated.
Parts (b) and (c) provide a much shorter proof.
[ii]
Follows directly from the property [iii].
[vii] The matrices
Let and commute. be a positive integer, then , and let . The three examples given at the end of this section are important. The first anticipates results in
section 2.6.
Example (Nilpotent matrices) A matrix N has nilpotency k if
For example, consider but . ,
where has nilpotency 2. Every matrix commutes with the identity matrix; in particular, the
matrices and N commute. Then from property (iii)
,
where only the first two terms in the series for are nonzero. Example (Roots of Identity). This result will be used in the next example. The matrix 11 has powers
,
recall the MacLaurin series , . Insert into the definition of the matrix exponential and To obtain
.
Example (Rotation Matrix). If is real and has a pair of complex conjugate eigenvalues, then
the associated eigenvectors will also be complex conjugate. Let the eigenvectors be denoted
and
. We will see in section 2.5 that when A is transformed using the
matrix
we get the form
. From the example above
.
The matr...
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 Fall '14
 MarcEvans

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