The text Basic Linear Algebra by Blyth and Robertson is a splendid introduction to the subject. It is published in the Springer SUMS series and the ISBN is 3-540-76122-5. This text gives a direct and short proof of the Cayley-Hamilton theorem, via some cunning matrix algebra. However, I have written some notes which get to the decomposition into generalized eigenspaces very fast using a road less travelled. You can read off the Cayley-Hamilton theorem and the characterization of diagonalizability in terms of the minimum polynomial at leisure. The formats are dvi, postscript and pdf.

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