Book suggestion for linear algebra "2"

I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been incredibly illuminating.

I would like to continue studying linear algebra, with particular focus on different properties of matrices and the transition to more general linear spaces (I am a physicist so Hilbert spaces and etc. are of particular interest).

Does anyone have a a good recommendation of books/resources/etc.?

I have to suggest the somewhat underrated Matrix Analysis by Horn and Johnson (the first edition was used for my ALA class at NCF.) They take a wonderfully concrete approach to most topics encountered in a second linear algebra course (Schur Decomposition, Spectral Theorem for Normal Operators, Jordan Canonical Form, Singular Value Decomposition) while adding a lot of other nice things into the mix. The fourth chapter on Hermitian Matrices talks about the Rayleigh Ritz Theorem and variational characterization of eigenvalues, which I imagine come up a lot in serious study of classical mechanics. Chapter five discusses finite dimensional inner product / normed / pre-normed spaces in terms of algebraic, analytic, and geometric properties. They include a discussion of completeness and the $l^p$ norms, which I guess could be seen as a preview of Hilbert Space Theory. There are also nice sections on the Gersgorin circle theorem and numerically solving linear systems.

I think it's a wonderful choice for any student, but especially a non-mathematician. The proofs are rigorous and sometimes tedious but always understandable. Typically, things are proved in an algorithmic fashion rather than through diagram chasing or algebraic artifice (nary a mention of finitely generated modules over a principal ideal domain.) My only complaint is that there are a fair number of results assumed regarding matrix algebra and determinants which wouldn't typically appear in a linear algebra course - references for these are typically not too hard to find though.

You could try Meyer's Matrix Analysis and Applied Linear Algebra or Lax's Linear Algebra and its Applications.

I took a "second semester" course in linear algebra out of Friedberg, Isnel and Spence's book Linear Algebra and found it to be a good read and a really useful reference.