SYMMETRICAL TOEPLITZ MATRICES WITH 2 PRESCRIBED EIGENPAIRS

The inverse problem of constructing a real symmetric Toeplitz matrix based on two prescribed eigenpairs is considered. Two new results are obtained. First, it is shown that the dimension of the subspace of Toeplitz matrices with two generically prescribed eigenvectors is independent of the size of the problem, and, in fact, is either two, three, or four, depending upon whether the eigenvectors are symmetric or skew-symmetric and whether n is even or odd. This result is quite notable in that when only one eigenvector is prescribed the dimension is known to be at least [(n + 1)/2]. Taking into account the prescribed eigenvalues, the authors then show how each unit vector in the null subspace of a certain matrix uniquely determines a Toeplitz matrix that satisfies the prescribed eigenpairs constraint. The cases where two prescribed eigenpairs uniquely determine a Toeplitz matrix are explicitly characterized.