I've been reading about matrix decompositions for a while now, but I always somehow seem to forget the reasons why they exist, and this irritates me when I see them applied in various branches of mathematics. Matrix decompositions I most often bump into are the SVD, LU, QR and the Woodbury matrix identity.
In this post, I would like to find out the intuition and why these decompositions exist, because understanding the why, is what illustrates to me the relevance and context of those decompositions, which helps me remember them better. In many source, these decompositions are simply stated, which I think, is analogous to writing uncommented C code. Sure you can write them down in one line and the assembler will compile them no problem, but it would be nice to see the motivation behind the authors writing this code.
My question thus is: How would one summarize each of these decompositions in couple of sentences, why were they derived? For what reason and application?
For the LU decomposition it helps determining for example the determinant of a matrix or its rank . Because the determinant of a triangular matrix is the product of its diagonal elements . Or calculating the inverse of a matrix (this method is used in programs).