random and generic matrices

random and generic matrices

Sections:

random matrices

matrices of variables

genericSymmetricMatrix

genericSkewMatrix

random matrices

To construct a random m by n matrix with entries in a ring R use the function random by typing random(R^m,R^n).

i1 : R = GF(3^2,Variable => a);

i2 : random(R^3,R^4)



o2 = | a+1 a    a  0   |

     | a-1 -a+1 1  0   |

     | a   -1   -a a+1 |



             3       4

o2 : Matrix R  <--- R

matrices of variables

To build an m by n matrix of variables drawn from the ring R, use genericMatrix. The syntax is genericMatrix(R,x,m,n) where R is the ring, x is the variable where we start and m and n specify the size of the matrix.

i3 : S = R[p..z];

i4 : genericMatrix(S,t,3,2)



o4 = | t w |

     | u x |

     | v y |



             3       2

o4 : Matrix S  <--- S

Note that to use the function genericMatrix the number of variables in the ring R must be at least as large as m*n.

genericSymmetricMatrix

To construct an n by n symmetric matrix whose entries on and above the diagonal are the variables of R use genericSymmetricMatrix. The syntax is genericSymmetricMatrix(R,x,n) where R is the ring, x is the variable you want to start with and n is the size of the matrix.

i5 : genericSymmetricMatrix(S,s,3)



o5 = | s t u |

     | t v w |

     | u w x |



             3       3

o5 : Matrix S  <--- S

genericSkewMatrix

To construct an n by n skew symmetric matrix whose entries above the diagonal are the variables of R use genericSkewMatrix. The syntax is genericSkewMatrix(R,x,n) where R is the ring, x is the variable you want to start with and n is the size of the matrix.

i6 : genericSymmetricMatrix(S,u,3)



o6 = | u v w |

     | v x y |

     | w y z |



             3       3

o6 : Matrix S  <--- S