<p>Sylvester's inertia law deals with the signatures of quadratic forms. I was thinking that it may be possible to extend this to multilinear forms; here is a first attempt.</p>

<p>Let $M$ be a $k$-linear form over the reals. For $1 \leq i \leq k$, define $\lambda_i(M)$ as the maximum dimension of a vector space $F$ such that</p>

<p>(*) $M(u_1,...,u_i,v_{i+1},...,v_k) \neq 0$ for every non-null $u_1,...,u_i \in F$, $v_{i+1},...,v_k \in F^{\bot}$.</p>

<p>If we define $\mu_i(M) = \lambda_i(M) - \lambda_{i-1}(M)$, is it true that these values form a partition of $n$? If so, can we formulate some "extended inertia law"? There might be some connection with Young tableaux but we probably don't require that for our needs.</p>

CC BY-SA 3.0
2014-03-22T15:31:27.077Z
It is an interesting question but I am not sure why you asked it in CSTheory. I think you are more likely to get an answer in Mathoverflow.
Sasho Nikolov
CC BY-SA 3.0
2014-03-22T20:45:49.923Z
OTOH, there are many cstheory questions in MO and no one recommends they post them here :)
Suresh Venkat
CC BY-SA 3.0
2014-03-23T07:32:31.310Z
The problem description is not at all clear. How is $n$ defined for example. And is all the coordinate of $M$ belongs to the same space so that $F$ can be well defined. A more precise formulation of the problem would be adding the condition that $M:R^n \times \dots \times R^n \to R$.
caozhu
<linear-algebra>
Extensions of Sylvester's inertia law?
NisaiVloot
# Extensions of Sylvester's inertia law?

(Asked by: username_0 on 2014-03-22)

Sylvester's inertia law deals with the signatures of quadratic forms. I was thinking that it may be possible to extend this to multilinear forms; here is a first attempt.

Let $M$ be a $k$-linear form over the reals. For $1 \leq i \leq k$, define $\lambda\_i(M)$ as the maximum dimension of a vector space $F$ such that

(\*) $M(u\_1,...,u\_i,v\_{i+1},...,v\_k) \neq 0$ for every non-null $u\_1,...,u\_i \in F$, $v\_{i+1},...,v\_k \in F^{\bot}$.

If we define $\mu\_i(M) = \lambda\_i(M) - \lambda\_{i-1}(M)$, is it true that these values form a partition of $n$? If so, can we formulate some "extended inertia law"? There might be some connection with Young tableaux but we probably don't require that for our needs.

**Tags:** linear-algebra
> **Comments:**
> **username_1**: It is an interesting question but I am not sure why you asked it in CSTheory. I think you are more likely to get an answer in Mathoverflow.  
> **username_2**: OTOH, there are many cstheory questions in MO and no one recommends they post them here :)  
> **username_3**: The problem description is not at all clear. How is $n$ defined for example. And is all the coordinate of $M$ belongs to the same space so that $F$ can be well defined. A more precise formulation of the problem would be adding the condition that $M:R^n \times \dots \times R^n \to R$.  

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## 0 answers

