A new algebra of Toeplitz-plus-Hankel matrices and applications

We introduce a new simultaneously diagonalizable real algebra A of symmetrical centrosymmetrical matrices having a Toeplitz-plus-Hankel structure. We give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors. An application is the construction of a symmetrical Toeplitz-plus-centrosymmetrical Hankel matrix of equal row sums having a prescribed real spectrum. This matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows. In particular, for the isospectral flow corresponding to the construction of a regular Toeplitz matrix having prescribed eigenvalues. Moreover, if A is a noise representation of an unknown matrix in A of rank k then we give a procedure to approximate A by a matrix in A of rank k.

[1]  F. Diele,et al.  Centrosymmetric isospectral flows and some inverse eigenvalue problems , 2003 .

[2]  R. Plemmons,et al.  Structured low rank approximation , 2003 .

[3]  A. Gersho.  Adaptive equalization of highly dispersive channels for data transmission , 1969 .

[4]  Fasma Diele,et al.  On Some Inverse Eigenvalue Problems with Toeplitz-Related Structure , 2004, SIAM J. Matrix Anal. Appl..

[5]  Salvatore D. Morgera,et al.  On the reducibility of centrosymmetric matrices — Applications in engineering problems , 1989 .

[6]  Moody T. Chu,et al.  Real-Valued, Low Rank, Circulant Approximation , 2003, SIAM J. Matrix Anal. Appl..

[7]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[8]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

[10]  J. Weaver.  Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors , 1985 .

[11]  C. T. Mullis,et al.  Inverse eigenvalue problem for real symmetric Toeplitz matrices , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[12]  G. Golub,et al.  Structured inverse eigenvalue problems , 2002, Acta Numerica.

[13]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[14]  John G. Proakis,et al.  An estimate of the upper bound on error probability for maximum-likelihood sequence estimation on channels having a finite-duration pulse response (Corresp.) , 1973, IEEE Trans. Inf. Theory.

[15]  O. Rojo,et al.  Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices , 2004 .

[16]  Alan L. Andrew,et al.  Eigenvectors of certain matrices , 1973 .