A book for self-study of matrix decompositions 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 ?
 A: 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.
A: As an intuitive-level introduction I recommend Gilbert Strang's freely available video lectures of the linear algebra course in MIT.
To establish theoretical ground I'd read:


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*Shaldon Axler - Linear Algebra Done Right


This book is comprehensive, accessible and it tackles theorems in a way that provides insight into the motivation behind them rather than hitting you with a series of steps that hardly contribute to the big picture.


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*Harry Dym - Linear Algebra in Action


A graduate level text. Teaches basic linear algebra and beyond, demonstrating the usefulness of the subject. I've attended Professor Dym's lectures personally and I consider him the greatest lecturer I had seen on the subject. He was clear, formal and fun to learn from.
A: I recommend "Matrix Analysis", 
Roger A. Horn, Charles R. Johnson
A: These two books


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*By Dr. Gilbert Strang Table of Contents and book link here

*By Gene H. Golub and Charles F. Van Loan from Amazon
These books have what you are looking for (and the reviews are good).
A: At an undergraduate level, I'm going to recommend you "Matrix Analysis and Applied Linear Algebra", by Carl Meyer, because I almost always end up reading it when I'm searching for concepts related to matrices.
You can preview the whole book here: http://matrixanalysis.com/DownloadChapters.html
A: (I've made an answer out of my original comment.)
Apart from the venerable Matrix Computations by Golub and Van Loan (which KV has already mentioned in his answer), there is the series of books by Pete Stewart, entitled Matrix Algorithms. The two volumes published so far can be seen here and here. In both volumes, the decompositional point of view is taken in, respectively, the solution of linear equations and eigenproblems.
On that note, you might also want to take a peek at this survey article, also from Stewart.
