A Second-Order Perturbation Expansion for the SVD

Let A be a rank-deficient matrix and let N be a matrix whose norm is small compared with that of A. The left singular vectors of A can be grouped into two matrices $U_1 $ and $U_2 $ whose columns provide orthonormal bases for the p-dimensional column space of A and for its $n - p$ dimensional orthogonal complement. The left singular vectors of $\tilde A = A + N$ can also be partitioned into the first p columns, $\tilde U_1 $, and the last $n - p$ columns $\tilde U_2 $. When analyzing a variety of signal processing algorithms, it is useful to know how different the spaces spanned by $U_1 $ and $\tilde U_1 $ (or $U_2 $ and $\tilde U_2 $) are. This question can be answered by developing a perturbation expansion for the subspace spanned by a set of singular vectors. A first-order expansion of this type has recently been developed and used to analyze the performance of direction-finding algorithms in array signal processing. In this paper, a new second-order expansion is derived and the result is illustrated with two examples.