Page 89 - Textos de Matemática Vol. 39

SPECTRALLY ARBITRARY FACTORIZATION: THE NONDEROGATORY CASE
CHARLES R. JOHNSON AND YULIN ZHANG
Dedicated to Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. It is known that a nonsingular, nonscalar, n-by-n complex matrix A may be factored as A = BC, in which the spectra of B and C are arbitrary, subject to det B det C = det A. We show further that B and C may be taken to be nonderogatory, even when the target spectra include repeated eigenvalues. This is a major step in a broader question of how arbitrary the Jordan forms of B and C may be, given their target spectra. In the process, a number of tools are developed, such as a special LU factorization under similarity, that may be of independent interest.
1. Introduction
In [2] it was shown that any nonsingular nonscalar matrix A ∈ Mn(C) may be factored as A = BC, so that B, C ∈ Mn(C) have arbitrary spectra, subject
n only to the obvious determinantal condition detA =
βi
n i=1
γi, in which
i=1
β1, β2, ..., βn are the eigenvalues of B and γ1, γ2, ..., γn are the eigenvalues of C
(repeats allowed). This fact has proven quite useful, and it is surprising that it was not known earlier; a slight generalization is proven in [1]. If B and C have repeated eigenvalues, no indication is given in [2] what sort of Jordan structure they may have, and unfortunately, the proof there is not easily adapted to further specify the Jordan structure.
The Jordan structure of B and C cannot generally be taken to be arbi-
trary. (Suppose that B and/or C have repeated eigenvalues and 1-by-1 Jordan
blocks in case n = 2, for example.) Thus a natural, deeper, question is to ask
for a given A and specified eigenvalues β1, β2, ..., βn for B and γ1, γ2, ..., γn for
n i=1
βiγi = detA, what the Jordan form for B and C may be? One might
C,
expect that nonderogatory Jordan structure (one Jordan block for each distinct
2000 Mathematics Subject Classification. 15A23.
Key words and phrases. exceptional, Jordan form, minor, nonderogatory, special LU fac- torization, total multiplicity.
Supported by Centro de Matem´atica da Universidade do Minho/FCT through the research program POCTI.
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