Bhatia’s Matrix Analysis, Chapter 1
I’ve been reading the first chapter of Bhatia’s “Matrix Analysis” to review (multi)linear algebra, and it’s been a pleasure. In just four pages or so, he proves the existence of QR decompositions and Schur decompositions, gives the Cholesky Decomposition as an exercise, proves the Singular Value Decomposition, and gives the Polar decomposition and its equivalence to the SVD as an exercise. The brevity of the proofs are the consequence of them being the most straightforward arguments I’ve seen for proving these decompositions. The exercises which occur inline with the text are challenging, but doable; as an examples
If
is a contraction, show that
All in all, this is a great chapter– I can’t wait to work through the problems at the end.