## Annals of Functional Analysis

### Principal angles and approximation for quaternionic projections

Terry A. Loring

#### Abstract

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real $C^{*}$-algebra generated by two projections.

#### Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 176-187.

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.afa/1396833512

Digital Object Identifier
doi:10.15352/afa/1396833512

Mathematical Reviews number (MathSciNet)
MR3192019

Zentralblatt MATH identifier
1297.15035

#### Citation

Loring, Terry A. Principal angles and approximation for quaternionic projections. Ann. Funct. Anal. 5 (2014), no. 2, 176--187. doi:10.15352/afa/1396833512. https://projecteuclid.org/euclid.afa/1396833512

#### References

• T. Ando, Unbounded or bounded idempotent operators in Hilbert space, Linear Algebra Appl. 438 (2011), no. 10, 3769–3775.
• S.N. Afriat, Orthogonal and oblique projectors and the characteristics of pairs of vector spaces, Proc. Cambridge Philos. Soc. 53 (1957), 800–816.
• P. Benner and D. Kressner, Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. II, ACM Trans. Math. Software 32 (2006), no. 2, 352–373.
• Å. Björck and G.H. Golub, Numerical methods for computing angles between linear subspaces, Math. Comp. 27 (1973), 579–594.
• L.G. Brown, The rectifiable metric on the set of closed subspaces of Hilbert space, Trans. Amer. Math. Soc. 337 (1993), no. 1, 279–289.
• J. Dixmier, Position relative de deux variétés linéaires fermées dans un espace de Hilbert, Revue Sci. 86 (1948), 387–399.
• F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140–156.
• A. Galántai and Cs. J. Heged\Hus, Jordan's principal angles in complex vector spaces, Numer. Linear Algebra Appl. 13 (2006), no. 7, 589–598.
• T. Giordano, A classification of approximately finite real $C^*$-algebras, J. Reine Angew. Math. 385 (1988), 161–194.
• P.R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389.
• M.B. Hastings and T.A. Loring, Topological insulators and $C\sp *$-algebras: Theory and numerical practice, Ann. Physics 326 (2011), no. 7, 1699–1759.
• N.J. Higham, Computing the polar decomposition–-with applications, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1160–1174.
• C. Jordan, Essai sur la géométrie à $n$ dimensions, Bull. Soc. Math. France 3 (1875), 103–174.
• H. Lin, Almost commuting selfadjoint matrices and applications, Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 193–233.
• T.A. Loring, Factorization of matrices of quaternions, Exposition. Math. 30 (2012), no. 3, 250–267.
• Code for testing the algorithms in this paper is available at the Lobo Vault, hosted by the University of New Mexico, repository.unm.edu/handle/1928/23492.
• G.K. Pedersen, Measure theory for $C^{\ast}$-algebras. II, Math. Scand. 22 (1968), 63–74.
• I. Raeburn and A.M. Sinclair, The $C^{\ast}$-algebra generated by two projections, School of Mathematics, University of New South Wales, 1989.
• A.P.W. Sørensen, Semiprojectivity and the geometry of graphs, Ph.D. thesis, University of Copenhagen, 2012, www.math.ku.dk/noter/filer.