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

We introduce a new simultaneously diagonalizable real algebra AA 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 AA is a noise representation of an unknown matrix in AA of rank kk then we give a procedure to approximate AA by a matrix in AA of rank kk.