# Questions tagged [multilinear-algebra]

Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.

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### Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
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### Why are these graphs coming from 9-dimensional alternating trilinear forms so symmetric?

Let $\phi(x,y,z)$ be an alternating trilinear form on a space $V$ over a field $K$. Let $u \in \mathbb{P}(V)$ be a projective point over $V$, then we say that the rank of $u$ is equal to the rank of ...
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### Is it possible to simplify the coefficient matrix for large values of $x$?

If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
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### Singular value decomposition for tensor

I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A_{i,j,k}$ that cannot be decomposed in the following ...
1 vote
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### Two unknowns: one vector, one scalar, one equation

I would like to know if this equation is solvable for $a$ and $\alpha$: \begin{equation} \Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b \end{equation} $\Sigma$ & $\Gamma$ ...
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### On comparing the nuclear norm with the Hilbert-Schmidt norm for symmetric tensors

I am interested in the special case of a symmetric tensor $T_{i_1,\ldots,i_k}$ of rank $k$, where each index, say $i_\kappa$, where $1 \leq \kappa \leq k$, runs from $1$ to $2$. The entries of such a ...
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### Maximal common isotropic subspace for a finite family of skewforms

Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F$ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
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### Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon. This invariant is discussed briefly in the seminal paper On the Areas ...
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### $O(n)$ Polynomial invariant of symmetric tensors

I apologize in advance if this question is not up to the level of research level questions on Math overflow. I am a complete outsider to invariant theory/representation theory and would like someone ...
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### Can the eigenvalues of a real symmetric tensor be complex?

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$. Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^...
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### Algebraic structure of the space of multiaffine maps

Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number. It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector ...
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### Is there an analog of polarization for skew-symmetric forms?

This question might be too lightweight here but on math.SE it did not receive any feedback since May 2, so... Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of ...
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### Subgroup of $PGL(n(n-1)/2, \mathbb K)$ preserving the grassmannian $Gr(2, n)$

How can we determine the subgroup of $PGL(\wedge^2 \bar{\mathbb Q}^n)$ which preserves the grassmamnnian $Gr(2, n)$ embedded as a projective variety in $\wedge^2(\bar{\mathbb Q}^n)$ via the Plucker ...
Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows. Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...
Assume an m-way tensor $\mathcal{Z}$. $\mathcal{Z}_{p_1 p_2 ... p_m} = 0$ if any different indices match and $\mathcal{Z}_{p_1 p_2 ... p_m} = 1$ otherwise. It is a symmetric tensor. Now if it is 2-...