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
det:n timesโKnรโฆรKnโK |
The determinant of a matrix ๐ is then defined to be det(๐1,โฆ,๐n), where ๐jโKn denotes the jth column of ๐.
Starting with the definition
det(๐1,โฆ,๐n)=โฯโSnsgn(ฯ)M1ฯ1M2ฯ2โฏMnฯn | (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 1.
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 Kn
as linear combinations of
๐1=(100โฎ0),๐2=(010โฎ0),โฆโ๐n=(000โฎ1), |
the standard basis of Kn. Let ๐ be an nรn matrix. The jth column is represented as โiMij๐i; whence using multilinearity
det(๐) | =det(โiMi1๐i,โiMi2๐i,โฆ,โiMin๐i) | ||
=nโi1,โฆ,in=1Mi11Mi22โฏMinndet(๐i1,๐i2,โฆ,๐in) |
The anti-symmetry assumption implies that the expressions
det(๐i1,๐i2,โฆ,๐in) vanish if any two of the
indices i1,โฆ,in coincide. If all n indices are distinct,
det(๐i1,๐i2,โฆ,๐in)=ยฑdet(๐1,โฆ,๐n), |
the sign in the above expression being
determined by the number of transpositions required to rearrange
the list (i1,โฆ,in) into the list (1,โฆ,n). The sign is
therefore the parity of the permutation
(i1,โฆ,in). Since we
also assume that
det(๐1,โฆ,๐n)=1, |
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 |