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Looking for a book/article with a lucid exposition of the matrix exponential, preferably including the case of infinite matrices. Basic properties especially, but also differential equations are of interest.

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What kind of infinite matrices? What topology do you want the exponential to converge in? –  Qiaochu Yuan Sep 13 '12 at 18:08
    
@QiaochuYuan: Real and complex matrices. Component-wise convergence. –  Evan Aad Sep 13 '12 at 18:10
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For finite dimensional matrices you are essentially talking about exponential maps on lie groups. So look up any elementary nook on lie algebras and lie groups (may be it is worthwhile to consider compact lie groups first). Off hand I can think of M.Artin's Algebra, Bump's Lie Groups. –  s.b Sep 13 '12 at 18:37

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There's an outstanding introduction to the matrix exponential in the classic analysis text by Kenneth Hoffman, Analysis in Euclidean Space. I reviewed the book for the MAA Online a few years ago and fell in love with the book in Dover reprint. It has a lot of original and remarkable features for an undergraduate analysis text and there's no reason not to have a copy now.

It absolutely stunned me that the matrix exponential features so prominently in that book and makes me wonder why so many undergraduate analysis texts-Rudin and Apostol included-completely skirt the subject? It stuns me because it strikes me that this is the natural place to learn about this function, that it shouldn't be postponed until either advanced differential equations or graduate differential geometry-which is where most students first learn about it. All you need is basic linear algebra to study it. It just struck me as a very strange decision. Then again,there are so many topics to cover in this critical course that are more essential, it probably just got tossed as too specialized and limited in range to include.

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Thanks, i'll check it out. I love your enthusiasm. I'm like you: i derive great pleasure from reading a well-written math textbook. –  Evan Aad Sep 14 '12 at 8:17

For infinite matrices, check out

1) R. G. Cooke, Infinite matrices and sequence spaces. London, Macmillan, 1950

2) Chapter 12 of P. Dienes' The Taylor Series

3) Mozyrska and Bartosiewicz, Dualities for linear control differential systems with infinite matrices

4) Mozyrska and Bartosiewicz, Observability of a class of linear dynamic infinite systems on time scales

5) Chapter 5 of Curtain‏ & Zwart‏'s An Introduction to Infinite-Dimensional Linear Systems Theory

6) This math exchange post.

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http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf

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Thanks, Robert. And thank you too, Amzoti and s.b, if you can "hear" me. I'd like to add that also books on Ordinary Differential Equations often introduce and discuss matrix exponentiation. Unfortunately, all the books/articles mentioned on this thread seem to limit their scope to finite dimensional matrices. –  Evan Aad Sep 14 '12 at 4:34

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