The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem—can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ν, μ, λ, this paper finds that the ratio α=(λ−ν)/(μ−ν), including infinity if μ = ν, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ν, μ and λ as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and α is too large or if n is odd and α is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.
On the inverse problem of constructing symmetric pentadiagonal toeplitz matrices from three largest eigenvalues
RAGNI, Stefania
2005
Abstract
The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem—can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ν, μ, λ, this paper finds that the ratio α=(λ−ν)/(μ−ν), including infinity if μ = ν, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ν, μ and λ as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and α is too large or if n is odd and α is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
