The geometric mean decomposition. (English) Zbl 1063.15009
Let \({\mathbb C}\) be the field of complex numbers. The matrix \({\mathbb H}\) in \({\mathbb C}^{m\times m}\) is said to have a geometric mean decomposition if \({\mathbb H}\)=QRP\(^\star\), where \({\mathbb P}\), \({\mathbb Q}\) are matrices with orthonormal columns and \({\mathbb R}\) is a real upper triangular matrix with diagonal elements all equal to the geometric mean of the positive singular values of \({\mathbb H}\).
The authors develop algorithms for computing the geometric mean decomposition of a matrix \({\mathbb H}\). The interest in this problem arose in signal processing.
The authors develop algorithms for computing the geometric mean decomposition of a matrix \({\mathbb H}\). The interest in this problem arose in signal processing.
Reviewer: Erich W. Ellers (Toronto)
MSC:
| 15A23 | Factorization of matrices |
| 65F25 | Orthogonalization in numerical linear algebra |
| 94A11 | Application of orthogonal and other special functions |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Keywords:
geometric mean decomposition; matrix factorization; unitary factorization; singular value decomposition; Schur decomposition; QR decomposition; MIMO systems; algorithms; signal processingReferences:
| [1] | Costa, M., Writing on dirty paper, IEEE Trans. Inf. Theory, 29, 439-441 (1983) · Zbl 0504.94027 |
| [2] | Foschini, G. J.; Gans, M. J., On limits of wireless communications in a fading environment when using multiple antennas, Wireless Person. Commun., 6, 311-335 (1998) |
| [3] | Ginis, G.; Cioffi, J. M., Vectored transmission for digital subscriber line systems, IEEE J. Select. Areas Commun., 20, 1085-1104 (2002) |
| [4] | Golub, G. H.; Kahan, W., Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2, 205-224 (1965) · Zbl 0194.18201 |
| [5] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996.), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009 |
| [6] | Hager, W. W., Applied Numerical Linear Algebra (1988.), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0665.65021 |
| [7] | Higham, N. J., Accuracy and Stability of Numerical Algorithms (1996.), SIAM: SIAM Philadelphia · Zbl 0847.65010 |
| [8] | Y. Jiang, W.W. Hager, J. Li, The generalized triangular decomposition, SIAM J. Matrix Anal. Appl., submitted for publication; Y. Jiang, W.W. Hager, J. Li, The generalized triangular decomposition, SIAM J. Matrix Anal. Appl., submitted for publication · Zbl 1135.15010 |
| [9] | Y. Jiang, J. Li, W.W. Hager, Joint transceiver design for MIMO systems using geometric mean decomposition, IEEE Trans. Signal Process., accepted for publication; Y. Jiang, J. Li, W.W. Hager, Joint transceiver design for MIMO systems using geometric mean decomposition, IEEE Trans. Signal Process., accepted for publication · Zbl 1370.94011 |
| [10] | Kosowski, P.; Smoktunowicz, A., On constructing unit triangular matrices with prescribed singular values, Computing, 64, 279-285 (2000) · Zbl 0958.65045 |
| [11] | Parlett, B. N., The Symmetric Eigenvalue Problem (1980.), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0431.65017 |
| [12] | Raleigh, G. G.; Cioffi, J. M., Spatial-temporal coding for wireless communication, IEEE Trans. Commun., 46, 357-366 (1998) |
| [13] | Telatar, I. E., Capacity of multi-antenna Gaussian channels, European Trans. Telecommun., 10, 585-595 (1999) |
| [14] | Trefethen, L. N.; Bau, D., Numerical Linear Algebra (1997.), SIAM: SIAM Philadelphia · Zbl 0874.65013 |
| [15] | Weyl, H., Inequalities between two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. USA, 35, 408-411 (1949) · Zbl 0032.38701 |
| [16] | Wilkinson, J. H.; Reinsch, C., Linear algebra, (Bauer, F. L., Handbook for Automatic Computation, vol. 2 (1971.), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0219.65001 |
| [17] | J.-K. Zhang, A. Kavčić, X. Ma, K.M. Wong, Design of unitary precoders for ISI channels, in: Proceedings IEEE International Conference on Acoustics Speech and Signal Processing, vol. 3, IEEE, Orlando, FL, 2002, pp. 2265-2268; J.-K. Zhang, A. Kavčić, X. Ma, K.M. Wong, Design of unitary precoders for ISI channels, in: Proceedings IEEE International Conference on Acoustics Speech and Signal Processing, vol. 3, IEEE, Orlando, FL, 2002, pp. 2265-2268 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
