Questions tagged [tensor-rank]

For questions about tensor-ranks.

The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix.

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Why is a linear transformation a $(1,1)$ tensor?

Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and $W$ are both vector spaces). I think it must be…
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Are matrices rank 2 tensors?

I know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular or something else? Some sources I read seem to…
HansHarhoff
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Tensor product and linear dependence of vectors

Let $V_1, \ldots, V_k$ be complex vector spaces. Given $k$ vectors $v_1 \in V_1, \ldots, v_k \in V_k$, we define that the tensor product $v_1 \otimes \ldots \otimes v_k$ has rank 1. For any tensor $T \in V_1 \otimes \ldots \otimes V_k$, the rank of…
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Is the rank of a Tensor different from the rank of a Matrix?

As far as I know, the rank of a matrix is the dimension of the vector space generated by columns. In NumPy notation, x = np.array([[1, 2], [2, 4]]) has a rank of one. np.linalg.matrix_rank(x) confirms that it is one. While studying the TensorFlow…
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Upper bound CP tensor rank

I have a question about CP tensor ranks. In the following, $\mathcal X \in \mathbb R^{n_1 \times n_2 \times n_3}$ is a third-order tensor of CP rank $R$, i.e., there exist vectors $a_i$, $b_i$ and $c_i$ for $i = 1, \ldots, R$ of appropriate…
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Finding a basis for symmetric $k$-tensors on $V$

We say a function is $k$-linear if it takes $k$ values as input and is linear with respect to each of them. For example, determinant is a $n$-linear function. (If the matrix is $n \times n$) A tensor is a function $T:V \times V\times V\times…
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Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much known about maps that preserve this tensor…
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Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is…
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Rank of tensors in terms of ranks of associated linear maps

Let $V$ be a vector space over a field $k$, let $w \in \otimes^l V$ be a tensors. We call $w$ a simple tensor if it can be written as $$ w=w_1 \otimes w_2 \otimes \ldots \otimes w_l, $$ where $w_i \in V$ for $i=1,\dots, l$. Simple tensors are also…
Alex
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Tensor Notation

I'm just starting to learn about tensors, and have a question. I'm looking at the statement $\Lambda_{\mu}\,^{\alpha}= \eta_{\mu\nu}\eta^{\alpha\beta}\Lambda^{\nu}\,_{\beta}$ What is the difference between $\Lambda_{\beta}\,^{\nu}$ and…
Sarah Jayne
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Is a $1^{\mathrm{st}}$ rank tensor identical to a vector?

As far as I know, we define vectors as elements of a vector space, then there is an isomorphism (by choosing a basis) from the vector space to tuples of components in some field, $\mathbb{F}$ say. Shouldn't it be, then, that $1^{\mathrm{st}}$ rank…
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Is the determinant a tensor?

I was reading Schutz's book on General Relativity. In it, he says that a(n) $M \choose N$ tensor is a linear function of $M$ one-forms and $N$ vectors into the real numbers. So does that mean the determinant of an $n \times n$ matrix is a $0 \choose…
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Intuitive understanding of 2-forms, (1,1)-tensors, and other fundamental objects of exterior algebra or tensor algebra

My background consists mostly of a good level in linear algebra, abstract algebra, undergrad calculus, topology & probability, and some working knowledge of geometric algebra and category theory. I'm currently learning differential geometry and…
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What is a rank-1 tensor? What is the meaning of rank in this context?

I feel like different sources use the term "rank" differently, which is perhaps leading to my confusion. When I think of rank I think of number of linearly independent columns/rows, number of pivots in RREF, etc... So a vector always has a rank of…
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Weird Notation for Trace of an Endomorphism

I am having some difficulty understanding a piece of notation from Riemannian Geometry: and Introduction to Curvature by John M. Lee. In Section 2 just under equation 2.3 Lee defines the trace operator which lowers the rank of a tensor by 2. He…
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