# Tagged Questions

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

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### Decomposable 3-form. [on hold]

Prove the following statement: A $3$-form $\omega^{3}=\displaystyle\sum_{i,j,k}{\omega_{ijk}\sigma_{i}\wedge\sigma_{j}\wedge\sigma_{k}}$ is decomposable if and only if ...
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### Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
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### degree of a multilinear polynomial with zeros at hamming weight $k$ points

$p(x_1,...,x_n)$ be a multilinear polynomial, where each $x_i$ is $\{0,1\}$. $p$ vanishes on all the points with hamming weight $k$ for some fixed $k \leq n$, and is non-zero everywhere else. Can we ...
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### $M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...
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### raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...
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### Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we havek^{(0)}Q = ...
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### Orbit closures of real symmetric bilinear form

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
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### Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
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### Orbit closures of symmetric bilinear form

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
### Followup to my previous question, $M = \text{Image}(u^\infty) \oplus \text{Ker}(u^\infty)$.
See my previous question here, Intersection of images and union of kernels. Let $A$ be a ring (not necessarily commutative), let $M$ be an $A$-module, and let $u: M \to M$ be an $A$-module ...