determinant as a multilinear mapping
Let ๐=(Mij) be an nรn matrix with entries in a field
K. The matrix ๐ is really the same thing as a list of n column
vectors of size n. Consequently, the determinant
operation
may be
regarded as a mapping
The determinant of a matrix is then defined to be where denotes the column of .
Starting with the definition
(1) |
the following properties are easily established:
-
1.
the determinant is multilinear;
-
2.
the determinant is anti-symmetric;
-
3.
the determinant of the identity matrix
is .
These three properties uniquely characterize the determinant, and indeed can โ some would say should โ 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 |
---|---|
Canonical name | DeterminantAsAMultilinearMapping |
Date of creation | 2013-03-22 13:09:17 |
Last modified on | 2013-03-22 13:09:17 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 5 |
Author | rmilson (146) |
Entry type | Theorem |
Classification | msc 15A15 |
Related topic | ExteriorAlgebra |