contract(Matrix,Matrix) -- contract a matrix by a matrix

Synopsis

Function: contract
Usage:

h = contract(m,n)
Inputs:
- m, a matrix, a map m : F <--- P between free modules of ranks f and p.
- n, a matrix, a map n : G <--- Q between free modules of ranks g and q.
Outputs:
- h, a matrix, the contraction of n by m, a matrix with the shape h : dual F ** G <--- dual P ** Q, whose entry in the slot h_g*i+j,q*k+l is the result of contracting n_j,l, by m_i,k

Description

The arguments can also be ring elements or vectors.

i1 : R = ZZ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : f = vars R ** vars R

o2 = | x2 xy xz xy y2 yz xz yz z2 |

             1      9
o2 : Matrix R  <-- R

i3 : contract(transpose vars R, f)

o3 = {1} | x y z y 0 0 z 0 0 |
     {1} | 0 x 0 x y z 0 z 0 |
     {1} | 0 0 x 0 0 y x y z |

             3      9
o3 : Matrix R  <-- R

i4 : contract(x, f)

o4 = | x y z y 0 0 z 0 0 |

             1      9
o4 : Matrix R  <-- R

i5 : contract(y, f)

o5 = | 0 x 0 x y z 0 z 0 |

             1      9
o5 : Matrix R  <-- R

i6 : contract(z, f)

o6 = | 0 0 x 0 0 y x y z |

             1      9
o6 : Matrix R  <-- R

This function is identical to diff(Matrix,Matrix), except that the multiplication by integers that occurs during differentiation is omitted.

Ways to use this method:

contract(Matrix,Matrix) -- contract a matrix by a matrix
contract(Matrix,Number)
contract(Matrix,RingElement)
contract(Matrix,Vector)
contract(Number,Matrix)
contract(Number,Number)
contract(Number,RingElement)
contract(Number,Vector)
contract(RingElement,Matrix)
contract(RingElement,Number)
contract(RingElement,RingElement)
contract(RingElement,Vector)
contract(Vector,Matrix)
contract(Vector,Number)
contract(Vector,RingElement)
contract(Vector,Vector)

contract(Matrix,Matrix) -- contract a matrix by a matrix

Synopsis

Description

See also

Ways to use this method: