File r37/lisp/csl/html/r37_0627.html artifact 2fa566d138 part of check-in e1a8550313

<A NAME=Ratjordan>

<TITLE>Ratjordan</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>RATJORDAN</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
The operator <em>ratjordan</em> computes the rational Jordan normal form R 
of a 
<A HREF=r37_0345.html>matrix</A> (A say). It returns {R,P,P^-1} where P*R*P^-1 =
 A. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>ratjordan</em>(&lt;matrix&gt;) 
<P>
<P>
&lt;matrix&gt; :- a square 
<A HREF=r37_0345.html>matrix</A>. 
<P>
<P>
<P>
Field Extensions: 
<P>
<P>
By default, calculations are performed in the rational numbers. To 
extend this field the <em>arnum</em> 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). 
See 
<A HREF=r37_0626.html>frobenius</A> for an example. 
<P>
<P>
Modular Arithmetic: 
<P>
<P>
<em>ratjordan</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. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
 a := mat((5,4*x),(2,x^2));

       [5  4*x]
       [      ]
  a := [    2 ]
       [2  x  ]



 ratjordan(a); 

    [0  x*( - 5*x + 8)]   [1  5]    [        -5 ]  
  { [                 ],  [    ],   [1     -----] }
    [        2        ]   [0  2]    [        2  ]
    [1      x  + 5    ]             [           ]
                                    [        1  ]
                                    [0     -----]
                                    [        2  ]


 on modular; 

 setmod 23; 

 a := mat((12,34),(56,78)); 

       [12  11]
  a := [      ]
       [10  9 ]



 ratjordan(a); 

    [15  0]   [16  8]   [1  21]
  { [     ],  [     ],  [     ]  }
    [0   6]   [19  4]   [1  4 ]



 on balanced_mod;

 ratjordan(a);

    [- 8  0]   [ - 7  8]   [1  - 2]
  { [      ],  [       ],  [      ]  }
    [ 0   6]   [ - 4  4]   [1   4 ]

</TT></PRE><P>