Selçuk Journal
of
Applied Mathematics
WinterSpring, 2003
Volume 4
Number 1
Research Center
of
Applied Mathematics

SJAM
WinterSpring 2003, Volume 4  Number 1

On location of the matrix spectrum inside an elipse*

Ayþe Bulgak^{1} , G. Demidenko^{2}, and I. Matveeva^{2}* 
^{1} Research Centre of Applied Mathematics,
Selçuk University, Konya, Turkey;
Email : abulgak@selcuk.edu.tr
^{2} Sobolev Institute of Mathematics,
SB RAS, 630090 Novosibirsk,
Russia;
Email: demidenk@math.nsc.ru,
Email: matveeva@math.nsc.ru
Received: July 3, 2002^{
} 
Summary
In the present article we consider the problem on location of the spectrum
of an arbitrary matrix $A$ inside the ellipse
$$
{\cal E} = \{\lambda \in C: \
\frac{(\Re\lambda)^2}{a^2} + \frac{(\Im\lambda)^2}{b^2} = 1\},
\quad a > b.
$$
One of the authors (see~[9]) established a connection of the
problem with solvability of the matrix equation
$$
H  \left(\frac{1}{2a^2} + \frac{1}{2b^2}\right)A^*HA 
\left(\frac{1}{4a^2}  \frac{1}{4b^2}\right)
(HA^2 + (A^*)^2H) = C.
$$
In this article we construct a Hermitian positive definite
solution $H$ to the equation in the form of a power series. We
prove that the norm $\H\$ characterizes an immersion depth of
eigenvalues of the matrix $A$ in the inside of the ellipse ${\cal
E}$. On the base of these results we propose an algorithm to
determine whether the spectrum of the matrix $A$ belongs to the
inside of the ellipse ${\cal E}$. 

Key
words
ellipse, matrix spectrum, matrix equations, matrix series,
algorithm

Mathematics Subject Classification (2000): 15A18, 15A24, 65F30
(*) The research was financially supported by the
Scientific and Technical Research Council of Turkey (TUBITAK) in the
framework of the NATOPC Fellowships Programme.

Article
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Article
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