determinant as a multilinear mapping
The determinant of a matrix is then defined to be where denotes the column of .
Starting with the definition
the following properties are easily established:
the determinant is multilinear;
the determinant is anti-symmetric;
the determinant of the identity matrix is .
These three properties uniquely characterize the determinant, and indeed can — some would say should — be used as the definition of the determinant operation.
Let us prove this. We proceed by representing elements of as linear combinations of
the standard basis of . Let be an matrix. The column is represented as ; whence using multilinearity
The anti-symmetry assumption implies that the expressions vanish if any two of the indices coincide. If all indices are distinct,
the sign in the above expression being determined by the number of transpositions required to rearrange the list into the list . The sign is therefore the parity of the permutation . Since we also assume that
we now recover the original definition (1).
|Title||determinant as a multilinear mapping|
|Date of creation||2013-03-22 13:09:17|
|Last modified on||2013-03-22 13:09:17|
|Last modified by||rmilson (146)|