Selšuk Journal of Applied Mathematics

 Selšuk Journal of
  Applied Mathematics

Winter-Spring, 2003
  Volume  4
  Number 1

Research Center of 
  Applied Mathematics

 SJAM Winter-Spring 2003, Volume 4 - Number 1

On location of the matrix spectrum inside an elipse*

Ay■e Bulgak1 , G. Demidenko2, and I. Matveeva2*
1 Research Centre of Applied Mathematics,
  Selšuk University, Konya, Turkey;
  Email :
2 Sobolev Institute of Mathematics,
  SB RAS, 630090 Novosibirsk, Russia;

Received: July 3, 2002

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 NATO-PC Fellowships Programme.
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