**AUTHORS:**Said Agoujil, Abdeslem Hafid Bentbib

**Download as PDF**

**ABSTRACT:**
This paper presents a symplectic J-SVD like decomposition of 2n-by-2m rectangular real matrix based
on symplectic reflectors. The idea for this approach was to use symplectic reflectors to first reduce the matrix to
J-bidiagonal form and then transform it to a diagonal form by using sequence of symplectic similarity transformations.
This was done in parallel with the Golub-Kahan-Reinsch method. This method allowed us to compute
eigenvalues for the skew-Hamiltonian matrix AJA.

**KEYWORDS:**
Singular value decomposition (SVD), Hamiltonian matrix, Skew-Hamiltonian matrix, Symplectic matrix,
Symplectic reflector

**REFERENCES:**

[1] S. Agoujil: Nouvelles methodes de factorisa- ´ tion pour des matrices structurees, PHD Thesis. ´ Faculte des Sciences et Techniques-Marrakech. ´ Departement de Math ´ e matiques et Informatique ´ (February 2008).

[2] S. Agoujil and A. H. Bentbib: On the reduction of Hamilotonian matrices to a Hamiltonian Jordan canonical form, Int. Jour. Math. Stat. (IJMS), 4 Spring (2009), 12–37.

[3] S. Agoujil and A. H. Bentbib: New symplectic transformation on C 2n×2 : Symplectic reflectors, Int. Jour. of Tomography and Statistics (IJTS), 11 Summer(2009), 99–117.

[4] S. Agoujil, A. H. Bentbib and A. Kanber: A Structured SVD-Like Decomposition. WSEAS TRANSACTIONS on MATHEMATICS, Issue 7, Volume 11, July 2012, .

[5] A.G. Akritas, G.I. Malaschinok, P.S. Vigglas: The SVD-Fundamental Theorem of Linear Algebra, Non Linear Analysis Modelling and Control,Vol 11 (2006), 123–136.

[6] M. Bassour and A. H. Bentbib: Factorization of RJR of skew-Hamiltonian matrix using its Hamiltonian square root, Int. Jour. of Tomography and Statistics (IJTS), 8 Springer(2011).

[7] A. H. Bentbib and A. Kanber: A method for solving Hamiltonian eigenvalue problem, Int. Jour. Math. Stat. (IJMS) 7 Winter(2010).

[8] C. Brezinski: Computational Aspects of Linear Control Numerical Methods and Algorithms. Springer, 2002.

[9] G. Golub, W. Kahan: Calculating the Singular Values and Pseudo-Inverse of Matrix, J. SIAM Numerical Analysis, Ser. B, Vol 2 N. 2 (1965) printed in U. S. A, 205–224.

[10] G. Golub and C. Reinsch: Singular Value Decomposition and Least Square Solutions, In J. H. Wilkinson and C. Reinsch, editors, Linear Algebra, volume II of Handbook for Automatic Computations, chapter I/10, 34–151. Springer Verlag, 1971.

[11] R.J. Duffin: The Rayleigh-Ritz method for dissipative or gyroscopic systems, Quart. Appl. Math., 18 (1960), 215–221.

[12] P. Lancaster: Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford, UK, (1966).

[13] F. Tisseur and K. Meerbergen: The quadratic eigenvalue problem, SIAM Review, 43 (2001), 235–286.

[14] N. E. Mastorakis, Positive singular Value Decomposition, Recent Advances in Signal Processing and Communication (dedicated to the father of Fuzzy Logic, L. Zadeh), WSEAS-Press, (1999), 717.

[15] N. E. Mastorakis, The singular Value Decomposition (SVD) in Tensors (Multidimensional Arrays) as an Optimization Problem. Solution via Genetic Algorithms and method of Nelder- Mead, Proceeding of the 6th WSEAS Int. Conf. on System Theory Scientific Computation, Elounda, Greece, August 21-23, (2006), 7 13.

[16] V.Mehrmann: The Autonomus Linear Quadratic Control Problem, Theory and Numerical Solution, Number 163 in Lecture Notes in Control and Information Sciences. Springer- Verlag, Heidelberg, July 1991.

[17] H. Xu: An SVD-like matrix decomposition and its applications, Linear Algebra and its Applications, 368 (2003), 1–24.

[18] H. Xu: A Numerical Method For Comuping An SVD-like matrix decomposition, SIAM journal on matrix analysis and applications, 26 (2005), 1058–1082.

[19] C. Van Loan: A Symplectic method for approximating all the eigenvalues of Hamiltonian matrix, Linear Alg. Appl., 61 (1984), 233–251.