| Popis: |
Let A be an n-by-n matrix and M ( x , y , z ) = z I n + x ℜ ( A ) + y ℑ ( A ) , where ℜ ( A ) = ( A + A ⁎ ) / 2 and ℑ ( A ) = ( A − A ⁎ ) / ( 2 i ) . The inverse numerical range problem seeks a unit vector x corresponding to a given point z of the numerical range of A satisfying z = x ⁎ A x . A kernel vector function ξ = ξ ( x , y , z ) of M ( x , y , z ) with point ( x , y , z ) on the curve F A ( x , y , z ) = det ( M ( x , y , z ) ) = 0 plays the role of the unit vector x for the inverse numerical range. The columns of the adjugate matrix L ( x , y , z ) = ( L j k ( x , y , z ) ) of M ( x , y , z ) are kernel vector functions of M ( x , y , z ) . We prove the Abel theorem on the intersections of the algebraic curves F A ( x , y , z ) = 0 and L j k ( x , y , z ) = 0 . A concrete numerical example is provided to verify the result using the Maple package algcurves. |
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