I'm working on this application where I get the spectral decomposition (O,D) of a matrix A for free (A is of full rank) but not A itself, and I need to only recover the last column of A. I was wondering if there was a more efficient way to do this than reconstructing the whole of A first?
last column of SPD matrix given it's spectral decomposition
eigensystem
Yes, for matrices A, B, the last column of matrix product A*B can be written as A*(the last column of B).  You can use this fact to get the last column of the reconstruction using only O(N^2).