Lecture2A

# An operator for which is said to be bounded example

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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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