## Symmetric and Skew-Symmetric forms

**Point of post: **In this post I discuss the equivalent of section 29 and the end of 30, in more detail than is done in Halmos’s book.

*Motivation*

In my the last post we discussed how one can extend the notion of bilinear forms; one of the main reasons being because we desired to extend the notion of a determinant to higher dimensions (in some intuitive sense). It turns out though that the generality to which we extended it is often (not always) too general for the level of linear algebra we are operating at. In particular, instead of studying when are any arbitrary, possibly distinct vector spaces over some field we instead intend to study where . In this case we denote by . The reason why we would want to consider this scenario is because when we restrict our attention to multilinear forms on the direct sum of the same vector space the resultant space has some beautiful interplay with . In other words, there is a very natural way in which interacts with the elements of . Our first indication of this was symmetric bilinear forms. Which, phrased slightly differently say that for every , which isn’t very profound considering there are only two elements of . But, we will be able to go further and say, in a precise manner yet to be said, that *acts *on so that there is a meaningful way to “multiply” elements of and elements of . This, is where all our work with permutations starts to come into play (though, to be quite honest I probably developed the theory a little further than it really needed to be).

*Remark: *Halmos defines something in his book either incorrectly, or ascribes to it a property which isn’t true (see below).

*‘s Action On *

Let be a finite dimensional vector space over . Then, for every and every there is a way to associate the two to create a new element of given by . For example, if

then . Indeed this is a -form since

So, with this we can state the main reason for restricting our attention to :

**Theorem: ***Let be a finite dimensional -space, then acts on via the action*

*where*

**Proof: **It suffices to prove that for all and

for all . So, the first of these is clear since

for all . For the second condition we merely note that

and since were arbitrary it follows that as required.

From hereon out we drop the in in lieu of .

We are now prepared to discuss two important classes of -forms whose definitions are precipitated by the above theorem. Namely the generalization of symmetric bilinear forms t* *and their counterparts skew-symmetric bilinear forms*. *More explicitly, we call *symmetric *if for all (equivalently that the isotropy subgroup generated by is the full group, for those who know the jargon) and we call *skew-symmetric *if for all .

Now, with this it’s clear that not all -forms are symmetric, nor are all skew-symmetric, in fact not all are either. That said, we will see that from each -linear form there is a natural way for one to build a symmetric and an skew-symmetric form. Namely:

**Theorem: ***Let be a finite dimensional -space and then the -form given by *

*is symmetric.*

**Proof: **Let be fixed but arbitrary and let . We claim that . To do this it suffices to show that

is a bijection. To prove injectivity we merely recall that and surjectivity follows from for all from where bijectivity follows( this is really just a statement of Cayley’s theorem for ). Thus, we see that

from where it follows by the arbitrariness of that is symmetric.

As would be expected there is a analogous way to naturally procure an skew-symmetric form from any -form , namely:

**Theorem: ***Let be a finite dimensional -space and . Then, the -form given by*

*is skew-symmetric.*

**Proof:** Let be arbitrary. Using the fact that as proven in the last theorem we see that

where we’ve made tacit use of the fact that (proof can be be found here) and that (since ). Thus, since was arbitrary the conclusion follows.

*Remark: * is for *antisymmetric *another name for skew-symmetric forms.

**References:**

1. Halmos, Paul R. ” *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

[…] in our last post we restricted our study of multilinear algebra to that of spaces of the form in the attempt to […]

Pingback by Alternating Forms « Abstract Nonsense | November 14, 2010 |

[…] let where is the “antisymmetrizer function” described in the post on skew-symmetric forms. Note then that if then (note that this no longer evaluates to zero on […]

Pingback by Halmos Section 29,30 and 31 « Abstract Nonsense | November 18, 2010 |

[…] Remark: If the notation seems unfamiliar, that’s because I created my notation. See here and here […]

Pingback by Halmos Sections 32 and 33: Linear Transformations and Transformations as Vectors (Pt. I) « Abstract Nonsense | November 22, 2010 |