making generic matrices

making generic matrices

Here a few commands for making various sorts of generic matrices. A generic matrix is one whose entries are independent variables from the ring, subject to certain relations.

We begin by making a ring with enough variables to accomodate the examples.

i1 : R = ZZ/101[a..z];

We can make a general generic matrix. We specify the ring, the starting variable and the dimensions.

i2 : genericMatrix(R,c,3,5)



o2 = {0} | c f i l o |

     {0} | d g j m p |

     {0} | e h k n q |



             3       5

o2 : Matrix R  <--- R

We can also make a skew symmetric matrix or a symmetric matrix.

i3 : R = ZZ/101[a..i];

i4 : genericSkewMatrix(R,c,3)



o4 = {0} | 0  c  d |

     {0} | -c 0  e |

     {0} | -d -e 0 |



             3       3

o4 : Matrix R  <--- R

i5 : gs = genericSymmetricMatrix(R,a,3)



o5 = {0} | a b c |

     {0} | b d e |

     {0} | c e f |



             3       3

o5 : Matrix R  <--- R

Suppose we need a random symmetric matrix of linear forms in three variables. We can use random and substitute to obtain it.

i6 : S = ZZ/101[x,y,z];

i7 : rn = random(S^1, S^{9:-1})



o7 = {0} | 42x-50y+39z 9x-15y-22z 50x+45y-29z -39x+30y+19z -38x+2y-4z -36x-16y-6z -32x+31y-32z -38x+31y+24z -42x-50y-41z |



             1       9

o7 : Matrix S  <--- S

i8 : substitute(gs, rn)



o8 = {0} | 42x-50y+39z 9x-15y-22z   50x+45y-29z |

     {0} | 9x-15y-22z  -39x+30y+19z -38x+2y-4z  |

     {0} | 50x+45y-29z -38x+2y-4z   -36x-16y-6z |



             3       3

o8 : Matrix S  <--- S