Classification of Multilinear Functions

Question

In An Introduction to Manifolds, Tu defines:

"Denote $V^{k} = V \times \dots \times V$ the Cartesian product of k copies of a real vector space V. A function $f: V^{k} \mapsto \mathbb{R}$ is k-linear if it is linear in each of its k arguments."

As one goes on, the symmetric and alternating k-linear functions are clearly important examples. Are they the only examples, or are there other k-linear functions?

My attempt at understanding this: imagine we only swap two elements (the first and last WLOG), leaving the rest in the same position. Let $f(v_1, \dots, v_n) = \lambda f(v_n, \dots, v_1)$ for some $\lambda \in \mathbb{R}, \lambda \notin {-1,0,1}$. Then, $f(v_1, \dots, v_n) = \lambda f(v_n, \dots, v_1) = \lambda^{2} f(v_1, \dots, v_n)$. Thus, $f(v_1, \dots, v_n) = 0$.

Is this logic in scaling by $\lambda$ sound, or am I missing something? Overall, I could not find much information on the classification of all k-linear functions, so I am trying to get information on that.

Answer 1

Let $g:V^k\to\mathbb R$ be symmetric, nonzero, and $h:V^k\to\mathbb R$ be alternating, also nonzero. Then $f:=g+h$ is neither symmetric, nor alternating, since

\begin{align}f(v_n,\dots,v_1)&=g(v_n,\dots,v_1)+h(v_n,\dots,v_1)\\ &=g(v_1,\dots,v_n)-h(v_1,\dots,v_n).\end{align}

Now if $f$ were symmetric, this would be equal to $f(v_1,\dots,v_n)=g(v_1,\dots,v_2)+h(v_1,\dots,v_n)$ for all possible values of $v_,\dots, v_n$. Rearranging we would get $h=-h$ for all arguments, which would mean $h=0$, a contradiction. In a similar way, $f$ being alternating would lead to $g=0$, another contradiction. So $f$ is neither symmetric, nor alternating.

For a simple example, consider $f:\mathbb R^2\times\mathbb R^2\to\mathbb R,~f(v,w)=v^t(I+A)w$, where $I$ is the identity matrix and $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. It has $f(v,v)=\vert v\vert^2\neq0$ for all $v\neq0$, so it isn't alternating. But $f(e_x,e_y)=f(e_y,e_x)=-1$, so it isn't symmetric, either.

However!

Every bilinear form can be decomposed into a sum of alternating and symmetric multilinear forms. Let $f_s(v,w):=\frac12(f(v,w)+f(w,v))$ and $f_a(v,w):=\frac12(f(v,w)-f(w,v))$. A simple calculation show that $f_s$ is symmetric, $f_a$ is alternating, and $f=f_s+f_a$. The summands are also called the symmtric and alternating parts of $f$.