×

zbMATH — the first resource for mathematics

Perturbation analysis of the canonical correlations of matrix pairs. (English) Zbl 0811.15011
A perturbation analysis of the canonical correlation of matrix pairs is presented. Absolute perturbation bounds for normwise as well as componentwise perturbations are established. Examples are presented to exhibit that small relative perturbation of a matrix pair does not necessarily imply small relative perturbation in its canonical correlations.
A class of matrix pairs for which good relative perturbation bounds can be derived under certain conditions is identified.
Reviewer: S.Sridhar (Madras)

MSC:
15A21 Canonical forms, reductions, classification
15A42 Inequalities involving eigenvalues and eigenvectors
15A45 Miscellaneous inequalities involving matrices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, T.W., An introduction to multivariate statistical analysis, (1984), Wiley New York · Zbl 0651.62041
[2] Barlow, J.; Demmel, J., Computing accurate eigensystems of scaled diagonally dominant matrices, () · Zbl 0725.65043
[3] Clark, D., Understanding canonical correlation analysis, ()
[4] Björck, Å; Golub, G.H., Numerical methods for computing angles between linear subspaces, Math. comp., 27, 579-594, (1973) · Zbl 0282.65031
[5] Björck, Å; Paige, C.C., Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm, SIAM J. on matrix anal. appl., 13, 176-190, (1992) · Zbl 0747.65026
[6] Björck, Å, Numerics of Gram-Schmidt orthogonalization, () · Zbl 0801.65039
[7] van der Burg, E., Nonlinear canonical correlation and some related technique, (1988), DSWO Press Leiden
[8] Chatterjee, S.; Hadi, A.S., Sensitivity analysis in linear regression, (1988), Wiley New York · Zbl 0648.62066
[9] Demmel, J.; Veselić, K., Jacobi’s method is more accurate than QR, SIAM J. matrix anal. appl., 13, 1204-1245, (1992) · Zbl 0759.65011
[10] Gittins, R., Canonical analysis: A review with applications in ecology, (1985), Springer-Verlag Berlin · Zbl 0576.62069
[11] Golub, G.H.; Van Loan, C.F., Matrix computations, (1989), Johns Hopkins U.P., Baltimore · Zbl 0733.65016
[12] Golub, G.H.; Zha, H., The canonical correlations and their numerical computation, (), to appear · Zbl 0823.65041
[13] Higham, N.J., Iterative refinement enhances the stability of QR decomposition methods for solving linear equations, Bit, 31, 447-468, (1991) · Zbl 0736.65016
[14] Hotelling, H., Relation between two sets of variates, Biometrika, 28, 322-377, (1936) · Zbl 0015.40705
[15] Israëls, A., Eigenvalue techniques for qualitative data, (1987), DSWO Press Leiden
[16] Jordan, C., Essai sur la géométrie à n dimensions, Bull. soc. math., 3, 103-174, (1875) · JFM 07.0457.01
[17] Krzanowski, H., Sensitivity of principal component analysis, J. roy. statist. soc., B46, 558-569, (1984)
[18] Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Quart. J. math., 11, 50-59, (1960) · Zbl 0105.01101
[19] Paige, C.C., A note on a result of Sun ji-guang: sensitivity of the CS and GSVD decomposition, SIAM J. numer. anal., 21, 186-191, (1984) · Zbl 0585.65028
[20] Rao, C.R., Separation theorems for singular values of matrices and their applications in multivariate analysis, J. multivariate anal., 9, 362-377, (1979) · Zbl 0445.62069
[21] Rao, C.R.; Yanai, H., General definition and decomposition of projectors and some applications to statistical problems, J. statist. plann. inference, 3, 1-17, (1979) · Zbl 0427.62046
[22] Radhakrishnan, R.; Kshirsagar, A.M., Influence functions for certain parameters in multivariate analysis, Comm. statist. ser. A, 10, 515-529, (1981) · Zbl 0505.62048
[23] Stewart, G.W.; Sun, G.-J., Matrix perturbation theory, (1990), Academic Boston
[24] Sun, J., Perturbation of angles between linear subspaces, J. comput. math., 5, 58-61, (1987) · Zbl 0632.15009
[25] J. Sun, On Condition Numbers of a Nondefective Multiple Eigenvalues, Technical Report, Dept. of Math., Linköping Univ. · Zbl 0754.15014
[26] Wedin, P.Å, On angles between subspaces, (), 263-285
[27] Wilkinson, J.H., The algebraic eigenvalue problem, (1965), Claredon Oxford, England · Zbl 0258.65037
[28] Zha, H., A componentwise perturbation analysis of the QR decomposition, SIAM J. matrix anal. appl., 14, 1124-1131, (1993) · Zbl 0787.65014
[29] Zha, H., The singular value decompositions: theory, algorithms and applications, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.