Joint Approximate Diagonalization of Symmetric Real Matrices of Order 2
DOI:
https://doi.org/10.5540/tema.2016.017.01.0113Keywords:
Joint approximate diagonalisation, eigenvectors, optimisation.Abstract
The problem of joint approximate diagonalization of symmetric real matrices is addressed. It is reduced to an optimization problem with the restriction that the matrix of the similarity transformation is orthogonal. Analytical solutions are derived for the case of matrices of order 2. The concepts of off-diagonalising vectors, matrix amplitude and partially complementary matrices are introduced. This leads to a geometrical interpretation of the joint approximate diagonalization in terms of eigenvectors and off-diagonalising vectors of the matrices. This should be helpful to deal with numerical and computational procedures involving large matrices.References
Laurent Albera, Anne Ferre ́ol, Pierre Comon & Pascal Chevalier. Blind Identification of Overcom- plete MixturEs of sources (BIOME). Linear Algebra and its Applications, 391 (2004), 3–30.
Howard Anton & Chris Rorres. Elementary Linear Algebra – Applications Version, volume 10 (John Wiley & Sons, 2010).
Adel Belouchrani, Karim Abed-Meraim, Jean-Franc ̧ois Cardoso & Eric Moulines. A blind source separation technique using second-order statistics. Signal Processing, IEEE Transactions on 45(2) (1997), 434–444.
Abdelwaheb Boudjellal, A. Mesloub, Karim Abed-Meraim & Adel Belouchrani. Separation of dependent autoregressive sources using joint matrix diagonalization. Signal Processing Letters, IEEE 22(8) (2015), 1180–1183.
Angelika Bunse-Gerstner, Ralph Byers & Volker Mehrmann. Numerical methods for simultaneous diagonalization. SIAM Journal on Matrix Analysis and Applications, 14(4) (1993), 927–949.
Augusto V. Cardona & Jose ́ V.P. de Oliveira. Soluc ̧a ̃o ELT AN para o problema de transporte com fonte. Trends in Applied and Computational Mathematics, 10(2) (2009), 125–134.
Augusto V. Cardona, R. Vasques & M.T. Vilhena. Uma nova versa ̃o do me ́todo LT AN. Trends in Applied and Computational Mathematics, 5(1) (2004), 49–54.
Jean-Franc ̧oisCardoso&AntoineSouloumiac.Jacobianglesforsimultaneousdiagonalization. SIAM Journal on Matrix Analysis and Applications, 17(1) (1996), 161–164.
Gilles Chabriel, Martin Kleinsteuber, Eric Moreau, Hao Shen, Petr Tichavsky` & Arie Yeredor. Joint matrices decompositions and blind source separation: A survey of methods, identification, and appli- cations. Signal Processing Magazine, IEEE, 31(3) (2014), 34–43.
Marco Congedo, Bijan Afsari, Alexandre Barachant & Maher Moakher, Approximate joint diagonal- ization and geometric mean of symmetric positive definite matrices. PloS one, 10(4) (2015).
Lieven De Lathauwer. A link between the canonical decomposition in multilinear algebra and simul- taneous matrix diagonalization. SIAM Journal on Matrix Analysis and Applications, 28(3) (2006), 642–666.
Klaus Glashoff and Michael M. Bronstein. Matrix commutators: their asymptotic metric properties and relation to approximate joint diagonalization. Linear Algebra and its Applications, 439(8) (2013), 2503–2513.
Franc ̧ois Gygi, Jean-Luc Fattebert & Eric Schwegler. Computation of Maximally Localized Wan- nier Functions using a simultaneous diagonalization algorithm. Computer Physics Communications, 155(1) (2003), 1–6.
Marcel Joho. Newton Method for Joint Approximate Diagonalization of Positive Definite Hermitian Matrices. SIAM Journal on Matrix Analysis and Applications, 30(3) (2008), 1205–1218.
Steven J. Leon. Linear Algebra with Applications, Eighth edition (Pearson, 2010).
S.I. McNeill & D.C. Zimmerman. A framework for blind modal identification using joint approxi-
mate diagonalization. Mechanical Systems and Signal Processing, 22(7) (2008), 1526–1548.
Anthony J. Pettofrezzo. Matrices and Transformations (Dover Publications, Inc., 1966).
Dinh Tuan Pham. Joint approximate diagonalization of positive definite Hermitian matrices. SIAM Journal on Matrix Analysis and Applications, 22(4) (2001), 1136–1152.
Petr Tichavsky` & Arie Yeredor. Fast approximate joint diagonalization incorporating weight matri- ces. Signal Processing, IEEE Transactions on 57(3) (2009), 878–891.
Roland Vollgraf & Klaus Obermayer. Quadratic optimization for simultaneous matrix diagonaliza- tion. IEEE Transactions on Signal Processing, 54(9) (2006), 3270–3278.
Arie Yeredor. Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation. Signal Processing, IEEE Transactions on 50(7) (2002), 1545–1553.
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