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<A NAME=Frobenius>

<TITLE>Frobenius</TITLE></A>
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<B>FROBENIUS</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
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The operator <em>frobenius</em> computes the <em>frobenius</em> normal form F of
 a 

<A HREF=r37_0345.html>matrix</A> (A say). It returns {F,P,P^-1} where P*F*P^-1 =
 A. 
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 <P> <H3> 
syntax: </H3>
<em>frobenius</em>(&lt;matrix&gt;) 
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<P>
&lt;matrix&gt; :- a square 
<A HREF=r37_0345.html>matrix</A>. 
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Field Extensions: 
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By default, calculations are performed in the rational numbers. To 
extend this field the 
<A HREF=r37_0634.html>arnum</A> package can be used. The package must 
first be loaded by load_package arnum;. The field can now be extended 
by using the defpoly command. For example, defpoly sqrt2**2-2; will 
extend the field to include the square root of 2 (now defined by sqrt2). 
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Modular Arithmetic: 
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<em>Frobenius</em>can also be calculated in a modular base. To do this 
first type on modular;. Then setmod p; (where p is a prime) will set 
the modular base of calculation to p. By further typing on balanced_mod 
the answer will appear using a symmetric modular representation. See 

<A HREF=r37_0627.html>ratjordan</A> for an example. 
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 <P> <H3> 
examples: </H3>
<P><PRE><TT>
 a := mat((x,x^2),(3,5*x)); 

       [    2 ]
       [x  x  ]
  a := [      ]
       [3  5*x]


 frobenius(a);

     [         2]    [1  x]    [       - x ]
  {  [0   - 2*x ],   [    ],   [1     -----]  }
     [          ]    [0  3]    [        3  ]
     [1    6*x  ]              [           ]
                               [        1  ]
                               [0      --- ]
                               [        3  ]


 load_package arnum;

 defpoly sqrt2**2-2;

 a := mat((sqrt2,5),(7*sqrt2,sqrt2));


       [ sqrt2     5  ]
  a := [              ]
       [7*sqrt2  sqrt2]



 frobenius(a); 

    [0  35*sqrt2 - 2]    [1   sqrt2 ]    [           1  ]
  { [               ],   [          ],   [1       - --- ]  }
    [1    2*sqrt2   ]    [1  7*sqrt2]    [           7  ]
                                         [              ]
                                         [     1        ]
                                         [0   ----*sqrt2]
                                         [     14       ]

</TT></PRE><P>