A fast algorithm for tracking the rank-r SVD-approximant Q(t)P(t)UT(t) of a sliding window data matrix X(t) of dimension L×N is introduced, where P(t) is a square-root power matrix of dimension r×r with r<min{L,N}. This algorithm is based on the unsymmetric Householder partial compressor and uses a reorthonormalizing Householder transformation for downdating. The concept is numerically self-stabilizing and requires no leakage. The dominant complexity is 4Lr+3Nr multiplications per time update which is the lower bound in complexity for an algorithm of this kind. Applications occur in the area of adaptive array processing and other forms of adaptive processing in finite duration subspaces. A complete algorithm summary is provided. Computer simulations illustrate the operation of the algorithm.