# Singular

#### D.3.1.24 colred

Procedure from library matrix.lib (see matrix_lib).

Usage:
colred(A[,e]); A matrix, e any type

Return:
- a matrix B, being the column reduced form of A, if colred is called with one argument.
(as far as this is possible over the polynomial ring; no division by polynomials)
- a list L of two matrices, such that L[1] = A * L[2] with L[1] the column-reduced form of A and L[2] the transformation matrix (if colred is called with two arguments).

Assume:
The entries of A are in the base field. It is not verified whether this assumption holds.

Note:
* This procedure is designed for teaching purposes mainly.
* It applies rowred to the transposed matrix. proc gauss_col should be faster.
* It should only be used with exact coefficient field (there is no pivoting) over the polynomial ring (ordering lp or dp).
* Parameters are allowed. Hence, if the entries of A are parameters the computation takes place over the field of rational functions.

Example:
 LIB "matrix.lib"; ring r=(0,a,b),(A,B,C),dp; matrix m[8][6]= 0, 2*C, 0, 0, 0, 0, 0, -4*C,a*A, 0, 0, 0, b*B, -A, 0, 0, 0, 0, -A, B, 0, 0, 0, 0, -4*C, 0, B, 2, 0, 0, 2*A, B, 0, 0, 0, 0, 0, 3*B, 0, 0, 2b, 0, 0, AB, 0, 2*A,A, 2a;""; ==> print(colred(m));""; ==> 0,0,0,0, 2*C, 0, ==> 0,0,0,0, -4*C,(a)*A, ==> 0,0,0,(b)*B,-A, 0, ==> 0,0,0,-A, B, 0, ==> 1,0,0,0, 0, 0, ==> 0,0,0,2*A, B, 0, ==> 0,1,0,0, 0, 0, ==> 0,0,1,0, 0, 0 ==> list L=colred(m,1); print(L[1]); ==> 0,0,0,0, 2*C, 0, ==> 0,0,0,0, -4*C,(a)*A, ==> 0,0,0,(b)*B,-A, 0, ==> 0,0,0,-A, B, 0, ==> 1,0,0,0, 0, 0, ==> 0,0,0,2*A, B, 0, ==> 0,1,0,0, 0, 0, ==> 0,0,1,0, 0, 0 print(L[2]); ==> 0, 0, 0, 1, 0, 0, ==> 0, 0, 0, 0, 1, 0, ==> 0, 0, 0, 0, 0, 1, ==> 1/2, 0, 0, 2*C, 0, -1/2*B, ==> 0, 1/(2b), 0, 0, -3/(2b)*B, 0, ==> -1/(2a)*A,-1/(4ab)*A,1/(2a),-2/(a)*AC,(-2b+3)/(4ab)*AB,1/(2a)*AB