On a second‐order expansion of the truncated singular subspace decomposition

Matrix subspace decompositions have numerous applications in science and technology. One of these matrix decompositions is the singular subspace decomposition. Signal processing, image processing, and discrete rank-deficient problems, to cite only few, constitute classes of practical problems where such decompositions play an important role. To begin with, let the real‡ matrix A 2 <n m have rank r 6 min.n;m/ and have a singular subspace decomposition

[1]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[2]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[3]  Zhengyuan Xu,et al.  Perturbation analysis for subspace decomposition with applications in subspace-based algorithms , 2002, IEEE Trans. Signal Process..

[4]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[5]  P. Wedin.  Perturbation bounds in connection with singular value decomposition , 1972 .

[6]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[7]  C. Paige,et al.  History and generality of the CS decomposition , 1994 .

[8]  G. Stewart.  On the Perturbation of Pseudo-Inverses, Projections and Linear Least Squares Problems , 1977 .

[9]  Richard J. Vaccaro,et al.  A Second-Order Perturbation Expansion for the SVD , 1994 .

[10]  Serge Gratton,et al.  Sensitivity and Conditioning of the Truncated Total Least Squares Solution , 2013, SIAM J. Matrix Anal. Appl..

[11]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[12]  G. Stewart.  Perturbation theory for the singular value decomposition , 1990 .

[13]  Serge Gratton,et al.  The Exact Condition Number of the Truncated Singular Value Solution of a Linear Ill-Posed Problem , 2014, SIAM J. Matrix Anal. Appl..

[14]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[15]  G. Stewart.  A second order perturbation expansion for small singular values , 1984 .

[16]  G. Stewart.  A note on the perturbation of singular values , 1979 .