Characterization of Alternating Multilinear Forms, and the Determinant
Point of post: In this post we give a characterization of alternating forms in terms of the values they take on any basis, and with this we characterize the determinant as the unique alternating multilinear form on the columns of a matrix such that where is the identity matrix.
So, in the last few posts we’ve gone from discussing -linear forms on , to discussing -forms on just so that we could discuss symmetric and skew-symmetric forms, to finally discussing alternating forms, all with the motivation to somehow “deal” with the determinant function. So, now that we’re done with these discussion it would seem criminal to not devote an entire thread to the repercussions have on the function . Thus, in this post we give a theorem which basically says that a skew-symmetric form is determined entirely by one value, (it’s value on some basis) and from this prove the oft stated “alternate definition” of the determinant function as the “unique function of the columns of a matrix which is linear in each column, skew-symmetric, and has .
Characterization of -forms
Now that we have dealt enough with alternating -forms to get a “feel” for them, we can safely state some obvious theorems, which trim away the fluff in the evaluation of such forms. For example, we noticed in the last post that when one expands an alternating -form via some basis that “a lot” of the terms disappear via the alternating quality of the form. We now formalize how may “a lot” is
Theorem: Let be a -dimensional space with basis then
Proof: Recall that by definition
A little thought shows that the right hand side can be rewritten as
where . But, clearly we may partition into two blocks, namely
recall though that is an injection if and only if it’s a bijection so that and thus . Thus, it follows that may be rewritten as
but notice though that by definition of an alternating -form that if is not an injection then so that the first term in vanishes. But, recalling that for we have that
and thus with both of these facts in mind we see that may be rewritten as
Theorem: Let be a -dimensional -space with basis , then for any there is a unique alternating -form for which .
Proof: We claim that the function
is such a function as described in the theorem. It’s evidently multilinear and , so it suffices to prove that it’s alternating. To do this we note that for any transposition we have that where, as always, . Thus, suppose that
then we can see using our previous observation with to see that
But, noting that and for all and making the observation that since only switches the two slots which are identical (and in particular the coefficients of are identical) we see that
we may conclude that is equal to
and since and were arbitrary the conclusion follows.
Now, uniqueness is clear by the previous theorem.
With this theorem under our belts and the last two theorems of the last post we may confidently state that:
Corollary: The function
is the unique function alternating -linear form on (where is some field) for which
And thus, the determinant is the unique alternating -linear forms on the columns of the elements of for which . Moreover,
if and only if forms a linearly dependent set.
1. 1. Halmos, Paul R. ” Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print