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);
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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
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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];
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i4 : random(T^3,T^{4:-1})
o4 = | (-a+1)x+ay -x+(a+1)y 0 -x-y |
| (a-1)x (-a-1)x-y (a+1)x+(-a+1)y -x-ay |
| ay (-a-1)x+ay (a+1)x+ay ax-y |
3 4
o4 : Matrix T <--- T
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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];
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i6 : genericMatrix(S,t,3,2)
o6 = | t w |
| u x |
| v y |
3 2
o6 : Matrix S <--- S
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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
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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 = | u v w |
| v x y |
| w y z |
3 3
o8 : Matrix S <--- S
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