Vol. 19, No. 3, 1966

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On a problem of O. Taussky

Bernard Werner Levinger and Richard Steven Varga

Vol. 19 (1966), No. 3, 473–487
Abstract

Recently, O. Taussky raised the following question. Given a nonnegative n×n matrix A = (ai,j), let ΩA be the set of all n × n complex matrices defined by

˚ΩA ≡ {B = (bi,j) | |bi,j| = ai,j for all 1 ≦ i,j ≦ n}. (1.1)

Then, defining the spectrum S(M) of an arbitrary set M of n × n matrices B as

S (M  ) ≡ {σ | det(σI − B ) = 0 for some B ∈ M },    (1.2)

what can be said in particular about S(ΩA)? It is not difficult to see that S(ΩA) consists of possibly one disk and a series of annular regions concentric about the origin, but our main result is a precise characterization of S(ΩA) in terms of the minimal Gerschgorin sets for A.

Mathematical Subject Classification
Primary: 15.25
Milestones
Received: 29 October 1964
Published: 1 December 1966
Authors
Bernard Werner Levinger
Richard Steven Varga