# Singular

#### D.15.20.11 grtranspose

Procedure from library `gradedModules.lib` (see gradedModules_lib).

Usage:

Return:

Purpose:

Note:
no reordering is performend by this procedure

Example:
 ```LIB "gradedModules.lib"; ring r=32003,(x,y,z),dp; module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M); ==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\ x. module N = grsyz( grtranspose( M ) ); grview(N); ==> Graded homomorphism: r(2) + r + r(-4)^2 <- r(2) + r + r(-4)^2, given by a\ diagonal matrix, with degrees: ==> ..1 ..2 ..3 ..4 .... ==> --- --- --- --- +... ==> -2 : 0 - - - |..1 ==> 0 : - 0 - - |..2 ==> 4 : - - 0 - |..3 ==> 4 : - - - 0 |..4 ==> === === === === ==> -2 0 4 4 module L = grtranspose(N); grview( L ); ==> Graded homomorphism: r(-2) + r + r(4)^2 <- r(-2) + r + r(4)^2, given by a\ diagonal matrix, with degrees: ==> ..1 ..2 ..3 ..4 .... ==> --- --- --- --- +... ==> 2 : 0 - - - |..1 ==> 0 : - 0 - - |..2 ==> -4 : - - 0 - |..3 ==> -4 : - - - 0 |..4 ==> === === === === ==> 2 0 -4 -4 module K = grsyz( L ); grview(K); ==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\ x. // Corner cases: 0 <- 0! module Z = grzero(); grview(Z); ==> Graded homomorphism: 0 <- 0, given by zero (0^2) matrix. grview( grtranspose( Z ) ); ==> Graded homomorphism: 0 <- 0, given by zero (0^2) matrix. // Corner cases: * <- 0 matrix M1[3][0]; module Z1 = grobj( M1, intvec(-1, 0, 1) ); grview(Z1); ==> Graded homomorphism: r(1) + r + r(-1) <- 0, given by zero (3 x 0) matrix. grview( grtranspose( Z1 ) ); ==> Graded homomorphism: 0 <- r(-1) + r + r(1), given by zero (0 x 3) matrix. // Corner cases: 0 <- * matrix M2[0][3]; module Z2 = grobj( M2, 0:0, intvec(-1, 0, 1) ); grview(Z2); ==> Graded homomorphism: 0 <- r(1) + r + r(-1), given by zero (0 x 3) matrix. grview( grtranspose( Z2 ) ); ==> Graded homomorphism: r(-1) + r + r(1) <- 0, given by zero (3 x 0) matrix. ```