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-1 a-1 |
| -a a+1 -a-1 a-1 |
| 0 a -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 = | -x+(-a-1)y -ax+y (a-1)x+(-a-1)y -ax+y |
| (a+1)x+(a-1)y -x+y (-a-1)x+(a-1)y ax+y |
| (a-1)x+(a-1)y (a-1)x+(-a+1)y -x+ay (-a-1)x+(a+1)y |
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 = | t w |
| u x |
| v y |
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 = | s t u |
| t v w |
| u w x |
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 : genericSkewMatrix(S,u,3)
o8 = | 0 u v |
| -u 0 w |
| -v -w 0 |
3 3
o8 : Matrix S <-- S
|