TITLE: A book for self-study of matrix decompositions
QUESTION [7 upvotes]: I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc').
Is there a good book for self-study of the subject ?
Note that I don't want to read about different decompositions but rather understand the proof for their existence and if there is an explanation for "where the decomposition came from" it will be fantastic. 
Any suggestions ?

REPLY [9 votes]: I would highly recommend "Numerical Linear Algebra" by Trefethen and Bau. It's got a great exposition of many matrix decompositions and how they are useful for numerical analysis (basically how to solve linear algebra problems in the real world). In particular it's got a great section on SVD, QR and Cholesky decompositions. This is a great way to get perspective on why some of these decompositions are so useful in the real world because with just abstract presentations it will make you wonder why all of them are necessary.