Double, borderline, and extraordinary eigenvalues of Kac–Murdock–Szegő matrices with a complex parameter

Abstract

For all sufficiently large complex ρ, and for arbitrary matrix dimension n, it is shown that the Kac-Murdock-Szegő matrix Kn(ρ) = ρ |j−k| n j,k=1 possesses exactly two eigenvalues whose magnitude is larger than n. We discuss a number of properties of the two "extraordinary" eigenvalues. Conditions are developed that, given n, allow us-without actually computing eigenvalues-to find all values ρ that give rise to eigenvalues of magnitude n, termed "borderline" eigenvalues. The aforementioned values of ρ form two closed curves in the complex-ρ plane. We describe these curves, which are n-dependent, in detail. An interesting borderline case arises when an eigenvalue of Kn(ρ) equals −n: apart from certain exceptional cases, this occurs if and only if the eigenvalue is a double one; and if and only if the point ρ is a cusp-like singularity of one of the two closed curves.Date: April 16, 2019. 2010 Mathematics Subject Classification. 15B05, 15A18, 65F15.

Eigenvalue bifurcations in Kac–Murdock–Szegő matrices with a complex parameter

“…7 below), for d > 0 our approximate formulas underestimate (overestimate) the larger (smaller) of the two bifurcated normalized eigenvalues. By Section 6.3 of [2], the large-|ρ| behaviors of the corresponding magnitudes are |ρ| n−1 /n and 1/n, respectively. These facts , for case corresponding to Fig.…”

“…This curve is a single-loop Cassini oval. But for all n ≥ 3, it is unique among the borderline curves B (k) n because it has no cusp-like singularities [2]. It is not of interest to us here.…”

“…This paper is a direct continuation of [1] and [2]. For ρ ∈ C and n ≥ 3, we deal with the Toeplitz matrix…”

“…An implicit equation for this level curve can be easily found from (1.2). The level curve can also be determined by the methods of [2], where it is called the type-2 borderline curve and is denoted by…”

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

“…Example In this example, we turn to Kac–Murdock–Szegö matrices whose applications are well known and their spectral properties have been investigated in many references 37‐40 . The generalized Kac–Murdock–Szegö matrices are finite sums of n × n matrices of the form 41 where Q is a monic polynomial with real coefficients, , , and .…”