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Unformatted text preview: act and the Euclidean norm is continuous. Therefore
attains a maximum value on
the sphere and
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
the exponential of an operator in the same way: converges. . We formally define  Lemma 2.1. If T is a bounded linear operator then
Proof. By  is as well. and the series of real numbers converges to . Therefore the series  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
[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.
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
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
. We will see in section 2.5 that when A is transformed using the
we get the form
. From the example above
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- Fall '14