^{}Department of Mathematics, Arak University, P.O. Box 38156-8-8349, Arak, Iran.

Abstract

Given four complex matrices $A$, $B$, $C$ and $D$ where $Ainmathbb{C}^{ntimes n}$ and $D inmathbb{C}^{mtimes m}$ and let the matrix $left( begin{array}{cc} A & B \ C & D \ end{array} right) $ be a normal matrix and assume that $lambda$ is a given complex number that is not eigenvalue of matrix $A$. We present a method to calculate the distance norm (with respect to 2-norm) from $D$ to the set of matrices $X in C^{m times m}$ such that, $lambda$ be a multiple eigenvalue of matrix $left( begin{array}{cc} A & B \ C & X end{array} right) $. We also find the nearest matrix $X$ to the matrix $D$.

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