Abstract
In this paper we propose a new algorithm for the joint eigenvalue decomposition of a set of real non-defective matrices. Our approach resorts to a Jacobi-like procedure based on polar matrix decomposition. We introduce a new criterion in this context for the optimization of the hyperbolic matrices, giving birth to an original algorithm called JDTM. This algorithm is described in detail and a comparison study with reference algorithms is performed. Comparison results show that our approach provides quicker and more accurate results in all the considered situations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
van der Veen, A.J., Ober, P.B., Deprettere, E.F.: Azimuth and elevation computation in high resolution DOA estimation. IEEE Trans. Signal Proc. 40, 1828–1832 (1992)
Lemma, A.N., van der Veen, A.J., Deprettere, E.F.: Analysis of joint angle-frequency estimation using ESPRIT. IEEE Trans. Signal Proc. 51, 1264–1283 (2003)
Haardt, M., Nossek, J.A.: Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieveal problems. IEEE Trans. Signal Proc. 46, 161–169 (1998)
Albera, L., Ferréol, A., Chevalier, P., Comon, P.: ICAR, a tool for blind source separation using fourth order statistics only. IEEE Trans. Signal Proc. 53(10-1), 3633–3643 (2005)
Roemer, F., Haardt, M.: A closed-form solution for multilinear PARAFAC decompositions. In: IEEE SAM 2008, pp. 487–491 (2008)
Bunse-Gerstner, A., Byers, R., Mehrmann, V.: Numerical Methods for Simultaneous Diagonalization. SIAM J. Matrix Anal. Applicat. 14 (4), 927–949
Yeredor, A.: Non-Orthogonal Joint Diagonalization in the Least-Squares Sense with Application in Blind Source Separation. IEEE Trans. Signal Proc. 50(7), 1545–1553 (2002)
Karfoul, A., Albera, L., Birot, G.: Blind underdetermined mixture identification by joint canonical decomposition of HO cumulants. IEEE Trans. Signal Proc. 58(2), 638–649 (2010)
Strobach, P.: Bi-iteration multiple invariance subspace tracking and adaptive ESPRIT. IEEE Trans. Signal Proc. 48, 442–456 (2000)
Fu, T., Gao, X.: Simultaneous Diagonalization with Similarity Transformation for Non-defective Matrices. In: IEEE ICASSP 2006, pp. 1137–1140 (2006)
Goldstine, H.H., Horwitz, L.P.: A procedure for the diagonalization of normal matrices. J. ACM 6(2), 176–195 (1959)
Eberlein, P.J.: A Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an arbitrary matrix. Journal of the Society for Industrial and Applied Mathematics 10(1), 74–88 (1962)
Ruhe, A.: On the quadratic convergence of a generalization of the Jacobi method to arbitrary matrices. BIT Numerical Mathematics 8, 210–231 (1968)
Souloumiac, A.: Nonorthogonal joint Diagonalization by Combining Givens and Hyperbolic Rotations. IEEE Trans. Signal Proc. 57(6), 2222–2231 (2009)
Iferroudjene, R., Abed Meraim, K., Belouchrani, A.: A New Jacobi-like Method for Joint Diagonalization of Arbitrary non-defective Matrices. Applied Mathematics and Computation 211, 363–373 (2009)
Cardoso, J.-F., Souloumiac, A.: Jacobi Angles for Simultaneous Diagonalization. SIAM Journal on Matrix Analysis and Applications 17(1), 161–164 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Luciani, X., Albera, L. (2010). Joint Eigenvalue Decomposition Using Polar Matrix Factorization. In: Vigneron, V., Zarzoso, V., Moreau, E., Gribonval, R., Vincent, E. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2010. Lecture Notes in Computer Science, vol 6365. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15995-4_69
Download citation
DOI: https://doi.org/10.1007/978-3-642-15995-4_69
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15994-7
Online ISBN: 978-3-642-15995-4
eBook Packages: Computer ScienceComputer Science (R0)