# Questions tagged [multilinear-algebra]

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

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### How can I decompose $x^2(x-y)+y^2(y-z)+z^2(z-x)$ into at least two factors?

Decompose the following expression into at least two factors, and if it does not decompose, you must prove with mathematical reasoning why it does not decompose (be sure to say with a mathematical ...
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### Unsure of construction of tensor product from bases

I've trouble understanding the definition of tensor products from the bases of the spaces which the operations is applied Given two vector spaces $V$ and $W$ over the same field, with bases $B_V$ ...
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### If I have two symmetric $k$-tensors $T$ and $S$ on a vector space $V$ such that $S(v,...,v) = T(v,...,v)$ for all $v$ how can I show that $S = T?$ [duplicate]

If I have two symmetric covariant $k$-tensors $T$ and $S$ on a vector space $V$ such that $S(v,...,v) = T(v,...,v)$ for all $v$ how can I show that $S = T?$ Apparently this question can be answered ...
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### Intrinsic definition of tensor, tensor density, pseudotensors, even and odd tensors

Motivation I am trying to figure out what the intrinsic definition of different variants of "tensors" are supposed to be. I managed to find the transformation rules for the coefficients of ...
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### Linear approximation using $2D$ Linear Interpolation

Probably it is a simple question. but even after several hours of research, I could not find anything relevant. I have a function $f(x,y) = xy$, with $x$ and $y$ that belong to bounded, continuous ...
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### complex valued real-linear functionals

Let $V$ be a vector space over $\mathbb{R}$ and $V^{*}$ denote its dual space. Why is $V^* \otimes \mathbb{C}$ the space of complex-valued real-linear functionals on $V$?
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### Are these R-homomorphisms necessary in this argument?

I just started to learn some multilinear algebra, that I lacked in my undergraduate program. I decided to do it with the book of Douglas Northcott. He defines what is going to be the solution for the ...
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### What sort of transformation is a 3d cubic tensor?

I was curious about tensors and higher dimensional matrices. I know what a tensor is. But I can find next to no readable information about what or how to actually work with them? In short, what kind ...
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### What is the relationship between: $\det(A + E[i,i]) = \det(A) + \det(A')$ and multilinearity?

Studying graph theory I came across the proof of Kirchof's theorem for maximal trees ("The number of generating trees of a graph $G$ is equal to the determinant of the reduced Laplacian Matrix of ...
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### Proving Basis of a type (1,1) Tensor

I am trying to teach myself tensor products, and I often see claims that the basis for a type (p,q) tensor is of the form: $e_{i_1}\otimes \ldots e_{i_q}\otimes e^{j_1}\otimes \ldots \otimes e^{j_p}$. ...
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### Classification of Multilinear Functions

In An Introduction to Manifolds, Tu defines: "Denote $V^{k} = V \times \dots \times V$ the Cartesian product of k copies of a real vector space V. A function $f: V^{k} \mapsto \mathbb{R}$ is k-...
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### Understanding non-commutativity of tensor product when tensors are interpreted as R-valued linear maps

In the textbook Geometric Control of Mechanical Systems by Francesco Bullo and Andrew D. Lewis, I find the following: I am struggling to reconcile the definition of the tensor product given here with ...
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### Finding the change of basis matrix for a type (0,1) tensor

I am considering a tensor (in particular, the electric field), defined by $$E_m = g_{ij}^k c_{k\ell}^{ij}S_{\ell m}$$ Ultimately, this means that the tensor E is a rank 1, type (0,1) tensor, ...
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### Bound symmetric tensor by its action on powers?

Let $A_{ijk}$ be a symmetric tensor (meaning that $A_{ijk}=A_{jik}=A_{kij}=\cdots$). Is there a bound of the form  |A_{ijk}| \leq C(d) \sup_{\|x\|_2=1} \left| \sum_{i,j,k=1}^d x_ix_jx_k A_{ijk}\...
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### A criteron for vanishing of Lie algebra cohomology

I am reading a criterion of Professor Serre for the vanishing of cohomology of Lie algebras in his paper "Sur les groupes de congruence des variétés abéliennes II". To prove this criterion, ...
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### Sum of decomposable p-vectors is decomposable in exterior algebra

I'm studying Chapter 16 on Multilinear Algebra of MacLane and Birkhoff's Algebra and I'm a little confused about using decomposable vectors. The question I haven't been able to solve is this: Given $V$...
I'm sorry for the long post but the this subject is confusing to me. Context: On one hand wiki talks about pseudovectors as if they are maps $\Phi:V^k \to V$ on the physical vector space with the ...
Let $S$ and $T$ be two arbitrary sets and consider the free vector space $C(S)$ and $C(T)$ generated respectively by $S$ and $T$. Show that $C(S \times T)$ is isomorphic to $C(S) \otimes C(T)$. I know ...