I read http://en.wikipedia.org/wiki/Triangular_matrix, and it says a matrix is triangularisable. But I did not find any where talking about how to do that.

E.g. how can I transform this matrix into a triangle matrix?


-2,2,3
-1,1,3
2,0,-1



A book fitting your description quite well is 

Multidimensional Real Analysis by Duistermaat and Kolk, a 2-volume set: Differentiation and Integration.

It has rigorous, slick proofs, is highly theoretical, but with lots of (advanced) examples and many, many exercises. Much attention is given to the Inverse and Implicit Function theorem, and submanifolds of $\mathbb{R}^n$. The book is used in a second-year course at Utrecht University. I have to admit that it was quite hard to read for me when I took the course. But it is great as a reference, and years later I still consult it now and then.

Another nice book is Loomis & Sternberg - Advanced Calculus (freely available from Sternberg's website.)
