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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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.
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.