exterior power of a matrix

Since the i-th exterior power is a functor, it applies to matrices as well as to modules.

i1 : R = ZZ[vars(0..19)]

o1 = R

o1 : PolynomialRing

i2 : ff = genericMatrix(R,4,5)

o2 = | a e i m q |
     | b f j n r |
     | c g k o s |
     | d h l p t |

             4      5
o2 : Matrix R  <-- R

i3 : exteriorPower (2,ff)

o3 = | -be+af -bi+aj -fi+ej -bm+an -fm+en -jm+in -bq+ar -fq+er -jq+ir -nq+mr
     | -ce+ag -ci+ak -gi+ek -cm+ao -gm+eo -km+io -cq+as -gq+es -kq+is -oq+ms
     | -cf+bg -cj+bk -gj+fk -cn+bo -gn+fo -kn+jo -cr+bs -gr+fs -kr+js -or+ns
     | -de+ah -di+al -hi+el -dm+ap -hm+ep -lm+ip -dq+at -hq+et -lq+it -pq+mt
     | -df+bh -dj+bl -hj+fl -dn+bp -hn+fp -ln+jp -dr+bt -hr+ft -lr+jt -pr+nt
     | -dg+ch -dk+cl -hk+gl -do+cp -ho+gp -lo+kp -ds+ct -hs+gt -ls+kt -ps+ot
     ------------------------------------------------------------------------
     |
     |
     |
     |
     |
     |

             6      10
o3 : Matrix R  <-- R

Note that each entry of in the above matrix is a 2 by 2 minor (the determinant of a 2 by 2 submatrix) of the matrix ff.

exterior power of a matrix

See also