Multilinear Function

• A Relation is said to be an -linear function (a.k.a. multilinear function) if is a linear function of each row of an Matrix when the remaining rows are held fixed. That is

  Whenever

• (Friedberg 4.3) Any Linear Combination linear functions is an linear function

• An -linear function is alternating if whenever two adjacent rows are identical.

• Let be an alternating -linear function. Then the following are true.

  • If is obtained by interchanging any two rows of then
  • If two rows of an matrix are identical, then

• (Friedberg 4.5) Let be an alternating -linear function on . For each matrix and each , define

  Where is the matrix obtained from by deleting the -th row and -th column.

  Then is an alternating -linear function on the matrices with entries from .

Links §

• Friedberg, Insel and Spence