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19 votes
3 answers
2k views

Best known bounds on tensor rank of matrix multiplication of 3×3 matrices

Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...
Brian Rushton's user avatar
7 votes
1 answer
231 views

Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$). Can we prove that the following problems are equivalent: $$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
jayki's user avatar
  • 135
6 votes
1 answer
621 views

Tensor matricizations and their decompositions

Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
qbit-'s user avatar
  • 71
5 votes
1 answer
473 views

higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
mathstudent42's user avatar
4 votes
1 answer
432 views

Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor

I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...
GA316's user avatar
  • 1,269
3 votes
0 answers
489 views

Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
Hyeonseo Yang's user avatar
2 votes
0 answers
77 views

Rank-1 decomposability of symmetric tensors

My question is about rank-1 decomposability of symmetric tensors over the reals. Let $v_1,\dots,v_n\in\mathbb{R}^d$ be vectors. Construct the object: $$ V=\sum_{j=1}^n \underbrace{v_j\otimes v_j\...
hookah's user avatar
  • 1,096
1 vote
0 answers
155 views

Some kind of product of two 2d tensors to create a 3d tensor?

I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays): given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
Min Wu's user avatar
  • 461
1 vote
0 answers
140 views

Adjacency matrix/tensor operations for graph sequences?

Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise. Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
Matthieu Latapy's user avatar
1 vote
0 answers
305 views

tensor/hypermatrix analogues of $GL(n,\mathbb{C})$?

Please excuse me if this question turns out to be incredibly silly for one reason or another. Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? What I'm mainly ...
Timothy Foo's user avatar
  • 1,075
0 votes
1 answer
234 views

Rank of matrices and secant varieties

Consider the Segre embedding $\mathbb{P}^n\times \mathbb{P}^n\rightarrow \mathbb{P}^N$, and let $S\subset\mathbb{P}^N$ be its image. Then $rank(Z)\leq k$ implies that $Z\in Sec_k(S)$. Moreover if $Z\...
user avatar
0 votes
0 answers
133 views

A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"

Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126." On page 125, at the end of the proof of Theorem 4.3, I abstract ...
Geom Zari's user avatar
0 votes
0 answers
199 views

Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?

Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by: $$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
SiXUlm's user avatar
  • 111