# Questions tagged [multilinear-algebra]

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

113 questions
Filter by
Sorted by
Tagged with
172 views

49 views

### Efficient way to express a symmetric tensor in terms of rank one elements

Let $$P(\mathbf{x})=P(x_1, \ldots, x_n)=x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}$$ be a homogenous polynomial of degree $k=k_1+\cdots+k_n$. It follows from a standard polarization identity (see for ...
196 views

### A symmetric bilinear form and a Plücker identity

It turns out that a special case of something I'm working on gives, as a corollary, a rather 19th-century-looking elementary statement about the rank of a certain symmetric matrix. I thought I would ...
170 views

### Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
342 views

### Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions: Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...
115 views

91 views

### Looking for a tractable algorithm or formula for the determinant of a tensor

It is possible to define the determinant of a tensor. We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map. If $T:\{1,....,n\}^m\to \mathbb C$ ...
62 views

93 views

### Sum of certain decomposable elements

Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are linearly independent, completely ...
727 views

### Tracing the word “form”

Today the word form can refer to (at least) three different kinds of mathematical object: A homogeneous polynomial. This was apparently started by Gauss (1801), renaming what others had called ...
438 views

### Waring rank vs tensor rank of symmetric tensors?

Suppose we work in an algebraically closed field. Then, do the Waring rank (symmetric tensor rank) and tensor rank of a symmetric tensor coincide in general? Recall that tensor rank is rank with ...
110 views

### Non alternative $k$-linear maps vanishing on $\sum x_i=0$

Assume that $V$ is a finite dimensional real vector space of dimension $n$. Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple ...
51 views

### Maximum number of matrices satisfying given rank conditions

Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that (i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$; (ii)...
323 views

277 views

### Subgroups of the tensor product $A\otimes A$

I have this problem about subgroups of the tensor product of an abelian group $A$ with itself which arises from a complete different setting. I fell into this question studying quandles and quandle ...
Let $\mathcal{G}$ be denote the set of all $3 \times 3$ real symmetric matrices and let $\mathcal{G}^+$ denote the set of all $3 \times 3$ positive semidefinite matrices (see definition). Let $S: \... 1answer 159 views ### Conditions on$\beta$under which the trace pairing restricted to$\mathfrak{so}(V,\beta)$is positive (negative) definite Let$V$be a finite dimensional vector space over$ \mathbb{R}$. Let \begin{equation} \left\langle\:,\:\right\rangle:\mbox{End}(V)\otimes\mbox{End}(V)\rightarrow \mathbb{R}\end{equation} denote the ... 0answers 75 views ### About a particular definition of a Hessian of a function of tuples of matrices Say I have a function$L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$i.e it takes a tuple of$n$matrices of different dimensions and computes a number from them. Then I see being defined a ... 1answer 144 views ### A question on surjectivity of a bilinear quadratic map Let$a=(a_0, a_1, ..., a_n )$,$b=(b_0, b_1, ..., b_n )$that belong to${\mathbb R}^{n+1}$. Define polynomials$f_a (t)=a_0 +a_1 t+ ... + a_n t^n$and$f_b (t)=b_0 +b_1 t+ ... + b_n t^n$and let$f_{...
Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ a_1\...