# $k$-linear form is the sum of a symmetric and a skew-symmetric one

This is a problem from the book "Finite Dimensional Vector Spaces" by Paul R. Halmos:

If $$w$$ a $$k$$-linear form, and if the characteristic of the underlying field of scalars is different from 2 (that is, if $$1+1 \ne 0$$), then $$w$$ is the sum of a symmetric $$k$$-linear form and a skew-symmetric one.

I know how to prove this for a bilinear form $$w$$. Let $$a = \dfrac{w + w^T}{2}$$ and $$b = \dfrac{w - w^T}{2}$$, then $$a$$ is symmetric and b is skew-symmetric. How can I prove this for any $$k$$-linear form?

$$k$$-linear form on $$V$$ is a multilinear form on $$V_1 \bigoplus \cdots \bigoplus V_k$$ where $$V_1 = \cdots = V_k = V$$

I'm thinking about making a symmetric form $$w_{sym} = \sum_{\pi \in S_k}{\pi w}$$ and skew-symmetric $$w_{skew} = \sum_{\pi \in S_k}{(\mathbb{sign} \pi) \pi w}$$ ...

• Does $k$ mean the underlying field? – Joppy Jun 26 at 13:42
• @joppy I've added definition of $k$-linear form – Andreo Jun 26 at 21:00
• I don’t think that this is true in general. For $k=3$, you should be able to find a form which is not a sum of a symmetric and an alternating form. – Joppy Jun 27 at 2:12
• This is an exercise from FDVS and as it is written in the preface, some statements in the exercises may be true or false. And if they are false reader should prove it or find a counter example. – Andreo Jun 27 at 4:28