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random and generic matrices

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   0  a+1  |
     | 0    a-1 -a a    |
     | -a-1 a+1 -a -a+1 |

             3       4
o2 : Matrix R  <--- R
Over a polynomial ring, this will select elements in the base ring or field. TO obtain a matrix of (say) linear polynomials, use
i3 : T = R[x,y];
i4 : random(T^3,T^{4:-1})

o4 = | -axa-1y -ax1y   1xa+1y    ax1y   |
     | 1xa-1y  a+1x    a+1x-a-1y -ax-ay |
     | -a-1x1y -a-1xay -a-1x-1y  axa+1y |

             3       4
o4 : Matrix T  <--- T

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.
i5 : S = R[p..z];
i6 : genericMatrix(S,t,3,2)

o6 = | 1t 1w |
     | 1u 1x |
     | 1v 1y |

             3       2
o6 : 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.
i7 : genericSymmetricMatrix(S,s,3)

o7 = | 1s 1t 1u |
     | 1t 1v 1w |
     | 1u 1w 1x |

             3       3
o7 : 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.
i8 : genericSymmetricMatrix(S,u,3)

o8 = | 1u 1v 1w |
     | 1v 1x 1y |
     | 1w 1y 1z |

             3       3
o8 : Matrix S  <--- S