I wrote my dissertation on a problem in matrix analysis and I found that I had to read from several different sources to understand the material. I don't know which is the best book on the subject but I would suggest reading Bhatia's book and Horn's book. In addition to this, I would suggest that the students read a short research paper such as "Almost Commuting Unitaries" by Exel and Loring. The paper is only three pages and only assumes knowledge of basic real analysis/complex analysis and basic linear algebra. Moreover, it gives the class a good example of the interplay between analysis and linear algebra.