A 2ND-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 U1 and U2 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 A = A + N can also be partitioned into the first p columns, U1, and the last n - p columns U2. When analyzing a variety of signal processing algorithms, it is useful to know how different the spaces spanned by U1 and U1 (or U2 and U2) 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.