We report on two closely related research projects. The work on fixed diagonal idempotent matrices is joint work with Julian Giol, Leonid Kovalev, Nga Nguyen and James Tener, and the work on dual pairs of frames is joint work with Jimmy Dilles, Julian Giol and Nga Nguyen. It should be noted that Tener was an REU undergraduate student in our program last summer, and his work led to our project on fixed diagonals. The others were mentors last summer. Our main result on idempotent matrices states that the set of nxn complex idempotent matrices with constant diagonal 1/2 is a pathwise connected set. We note that the analogous statement for projections (i.e. the selfadjoint case) is an open question that has been studied by several researchers and is apparently very hard. There may be relations to the Kadison-Singer problem in the equivalent form of Anderson's Paving Problem. This provided some motivation for expanding the scope of the problem to idempotents. More motivation came from the work on dual pairs of frames. In a general (not necessarily finite-dimensional) setting, dual pairs of frames are in one-to-one correspondence with the unitarily equivalence classes of idempotents, just as Parseval frames (tight frames of frame bound 1) are in one-to-one correspondence with the unitarily equivalence classes of projections. So connectivity results for idempotents yield similar results for dual frame pairs, and this is the theme.