subspacea - Angles between subspaces

Version 4.6 (11.9 KB) by Andrew Knyazev

Angles between subspaces and canonical correlations in general scalar products.

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Updated 20 Dec 2021

subspacea(F,G,A) Finds all min(size(orth(F),2),size(orth(G),2)) principal angles between two subspaces spanned by the columns of matrices F and G in the A-based scalar product x'*A*y, where A is Hermitian and positive definite. COS of principal angles is called canonical correlations in statistics.

The input A can be provided as a matrix as well as a string of a function name, e.g., funcA giving the product funcA(X) = A*X.

[theta,U,V] = subspacea(F,G,A) also computes left and right principal (canonical) vectors - columns of U and V, respectively.

If F and G are vectors of unit length and A=I, the angle is ACOS(F'*G) in exact arithmetic. If A is not provided as a third argument, than A=I and the function gives the same largest angle as SUBSPACE.m by Andrew Knyazev. MATLAB's SUBSPACE.m function is still badly designed and fails to compute some angles accurately.

The algorithm is described in A. V. Knyazev and M. E. Argentati, Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates. SIAM J. Scientific Computing, 23 (2002), no. 6, 2009-2041. http://dx.doi.org/10.1137/S1064827500377332

Cite As

Andrew Knyazev (2024). subspacea - Angles between subspaces (https://www.mathworks.com/matlabcentral/fileexchange/55-subspacea-angles-between-subspaces), MATLAB Central File Exchange. Retrieved September 7, 2024.

Knyazev, Andrew V., and Merico E. Argentati. “Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates.” SIAM Journal on Scientific Computing, vol. 23, no. 6, Society for Industrial & Applied Mathematics (SIAM), Jan. 2002, pp. 2008–40, doi:10.1137/s1064827500377332.

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Created with R2021a

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Acknowledgements

Inspired by: subspace.m, ortha.m

Inspired: subspace.m

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subspacea

Version	Published	Release Notes
4.6	20 Dec 2021	Fix error handling, code beautification.	Toolbox Zip
1.2	15 May 2015	Updated description. added a conversion to a toolbox	Toolbox Zip
1.1.0.0	19 May 2009	License update to free software (BSD). Comments update.	Download
1.0.0.0	28 Jun 2005	minor update for MATLAB R14	Download

MLA	Knyazev, Andrew V., and Merico E. Argentati. “Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates.” SIAM Journal on Scientific Computing, vol. 23, no. 6, Society for Industrial & Applied Mathematics (SIAM), Jan. 2002, pp. 2008–40, doi:10.1137/s1064827500377332.
APA	Knyazev, A. V., & Argentati, M. E. (2002). Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates. SIAM Journal on Scientific Computing, 23(6), 2008–2040. Society for Industrial & Applied Mathematics (SIAM). Retrieved from https://doi.org/10.1137%2Fs1064827500377332
BibTeX	@article{Knyazev_2002, doi = {10.1137/s1064827500377332}, url = {https://doi.org/10.1137%2Fs1064827500377332}, year = 2002, month = {jan}, publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})}, volume = {23}, number = {6}, pages = {2008--2040}, author = {Andrew V. Knyazev and Merico E. Argentati}, title = {Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates}, journal = {{SIAM} Journal on Scientific Computing} }