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

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4
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1answer
46 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
2
votes
1answer
56 views

Kernel of the Symmetrizing Map $Sym:\bigotimes^k V\to \bigotimes^k V$

$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ ...
-1
votes
0answers
18 views

reduction to canonical form of the quadratic form that corresponding to the matrix [on hold]

Do reduction to canonical form of the quadratic form that corresponding to the matrix: $$\begin{bmatrix} 1 & -1 & 0 \\ -1 & -2 & -1 \\ 0 & -1 & 2 \\ \end{bmatrix}$$ $$\in ...
1
vote
0answers
20 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
0
votes
1answer
28 views

Tensor Product of Vectors

let $S,T$ be respectively $k-, n- $ tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,....,x_{k+n}):=T(x_1,...,x_k)S(x_{k+1},...,x_{k+n}) $$ (their product as Real numbers, ...
0
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0answers
22 views

Definition of Tensors Over Complex Numbers

My question is two part. First, how does the definition of tensors and tensor spaces change when the vectors that the tensors act upon are elements of a complex vector space as apposed to when they ...
2
votes
0answers
15 views

Mahalanobis distance in $\mathbb R^3$ or more dimension

Does anyone know how Mahalanobis distance looks like in $\mathbb R^3$ or $n$ dimensions. I would also be grateful if someone could give a geometrical interpretation in $\mathbb R^3$? Thank you a lot.
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0answers
21 views

More on the $n$-dimensional cross product: Orientation

Wikipedia states: This formula is identical in structure to the determinant formula for the normal cross product in $\mathbb R^3$ except that the row of basis vectors is the last row in the ...
1
vote
2answers
27 views

What is the right hand side in this definition of $n$-dimensional cross product

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one ...
0
votes
2answers
27 views

Tensor manipulation

I am very new at manipulating tensors and I have the following equation: $$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$ where $\tau$ is the independent index and $g_{\tau \rho}$ the metric ...
1
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0answers
21 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
2
votes
0answers
34 views

Tensor transpose notation

I have a rank 3 tensor $\mathbf{Q}$. What notation should I use to denote the transposition of two of the dimensions? For instance, if I want to transpose the first and second dimensions, one way I ...
4
votes
2answers
56 views

Linear Maps as Tensors

Let $V$ and $W$ be finite dimensional vector spaces and let $V^{\ast}$ denote the dual $V$. I read that the space $V^{\ast}\otimes W$ may be thought of in four different ways: as the space of linear ...
3
votes
1answer
21 views

Tensor, exterior, symmetric powers over fields of nonzero characteristic

I was reading Fulton and Harris' discussion of exterior and symmetric products as quotient spaces of tensor products in their rep theory book when I noticed that they made this claim (the emphasis is ...
0
votes
0answers
23 views

What's the name of this tensor product?

Fix $V$ to be a vector space over $\mathbb{R}$. For all $k \in \mathbb{N}$, let $L_k$ be the space of all $k$-tensors on $V$, and let $S_k$ be the set of all permutations of the set $\{1,\dots, k\}$. ...
2
votes
1answer
63 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
1
vote
1answer
34 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
1
vote
1answer
40 views

If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
0
votes
1answer
35 views

Representing a series of Matrix inner product with a single matrix product.

I have a set of constraints in my optimization problem, constraints in the form , $\langle A, e_i e_j^T \rangle = r_{ij} ,\forall i,j \epsilon S$, where $A$ is an $n*n$ semidefinite and symmetric ...
4
votes
1answer
85 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
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vote
0answers
15 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
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votes
1answer
28 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't ...
4
votes
0answers
53 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
-1
votes
1answer
25 views

What is the best fit (in the sense of least-squares) to the data?

A) Find the best fit (in the sense of least-squares) to the data $x_1$ $(1,-1,-1,1)$ $x_2$ $(1,1,-1,-1)$ $y$ $(5,1,1,1)$ by a linear function of the form $y$=$a$+$bx_1$+$cx_2$ B) Find ...
8
votes
2answers
38 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
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vote
2answers
38 views

Contraction as Adjoint of Wedging

Let $V$ be an $n$-dimensional vector space. Given $\phi^1\wedge \cdots \wedge \phi^k\in \bigwedge ^k(V^*)$ and $v_1\wedge\cdots\wedge v_k\in \bigwedge^k(V)$, we write $$ \langle \phi^1\wedge ...
7
votes
1answer
213 views

Symmetric kernel of tensor product

Let $V,W$ be two real vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with distinct kernels $K_i$ of dimension $1$. Consider the tensor product of these maps ...
3
votes
1answer
47 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
0
votes
1answer
47 views

What operation is “$\oplus$” in Lounesto's introduction to Clifford Algebras

I'm reading Lounesto's CLifford Algebras and Spinors and on page 26 (also below) he states the following: \begin{align} C\mathcal{l}_2=\mathbb{R}\oplus\mathbb{R}^2\oplus\bigwedge^2\mathbb{R}^2. ...
0
votes
1answer
52 views

Complex functions and $\star_3$

I wanted to maybe extend Hodge star/ Technical question to a new question so others could benefit from the idea. So there we discussed that when the $\star$ is Hodge duality star then it is ...
0
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0answers
25 views

Intrinsic Definition of $^\sigma\alpha$, where $\alpha$ a covariant $k$-tensor and $\sigma\in S_k$

Let $V$ be a finite dimensional vector space and $\mathcal T^k(V^*)$ be the set of all the covariant $k$-tensors on $V$. The symmetric group $S_k$ acts on $\mathcal T^k(V^*)$ as follows: Given ...
0
votes
1answer
40 views

What is the prerequisiste to study Tensors for application in signal processing?

I want to study Blind Source separation in signal processing for this I need to study Tensors and have a basic idea about rank, border rank and other concepts. Right now I am studying from ...
1
vote
1answer
30 views

Some detail needed in Positive Definite Matrix

First all of all, I am sorry I have put a page of my text book. Somehow I need some help to understand some paragraphs in the page. If you can explain, please let me have some explanations to those in ...
2
votes
3answers
42 views

Linear Algebra Question for Repeated Eigenvalues

Can somebody explain what the sentence in the red circle means?
0
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0answers
24 views

use universal properties to prove the existence of isomorphism

Use universal properties to prove that for a finite dimesional vector space $V$ and $W$ there is a canonical isomorphism: $$\bigwedge^2(V\oplus W)\to \bigwedge^2V\oplus(V\otimes W)\oplus\bigwedge^2W$$ ...
1
vote
1answer
59 views

How to prove that below quantity is a Third Rank Tensor

$F^{ik}$ is an antisymmetric tensor. I want to prove that below quantity is a Third Rank Tensor. $$\dfrac{\partial F_{ik}}{\partial x^{l}} + \dfrac{\partial F_{kl}}{\partial x^{i}} + \dfrac{\partial ...
2
votes
0answers
34 views

$L\cdot M:=\pi_{r+s}(L\otimes M)$, then $((f_i\cdot f_j)\cdot f_k)\cdot f_l=?$

I'm reading Kenneth Hoffman's "Linear Algebra", Ed 2. In $\S5.7$ "the Grassman Ring" it tries to explain the way lead to the definition of exterior product (a.k.a. wedge product). However I got some ...
0
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0answers
19 views

on “strong positive semi-definiteness”

Suppose $M_{ij}^{kl}$ is a real symmetric $nm \times nm$ matrix ($0 \leq i,j \leq n$ and $0 \leq k,l \leq m$), i.e. $\forall i,j,k,l$ \begin{equation} M_{ij}^{kl} = M_{ji}^{lk}. \end{equation} Can we ...
2
votes
1answer
27 views

From linear transformation to alternating linear transformation

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.6$ "Multilinear Functions" it talks about generating an alternating linear transformation from a linear transformation. The collection ...
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0answers
37 views

Why are free objects related to tensors important?

Let $R$ be a commutative ring and $M$ be an $R$-module. Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any ...
3
votes
1answer
52 views

Is the symmetric algebra direct sum of $k$-th symmetric powers?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $S(M)$ be the symmetric algebra of $M$ and $S^n(M)$ be $n$-th symmetric power of $M$. Then is $S(M)$ the internal direct sum of $S^n(M)$? ...
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0answers
39 views

What do we call $R$-bilinear maps $V,W \rightarrow R$?

Let $R$ denote a fixed commutative ring. Then given an $m \times n$ matrix $A$ with entries in $R$, we get an $R$-linear transform $R^n \rightarrow R^m$ in the usual way. We also get an $R$-bilinear ...
4
votes
1answer
57 views

Plücker Relation: misunderstanding?

I'm trying to understand exterior algebra better by gaining some "bare hands" understanding of the exterior powers $\Lambda^k(X)$ in more detail when $\dim(X)$ is small. I think so far I understand ...
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0answers
36 views

Tensor Invariants

I was learning about tensors and vectors and it was mentioned in class that they were "invariant" objects that take on specific representations in different bases. I get this for vectors -- I can ...
8
votes
3answers
115 views

Is it true that every element of $V \otimes W$ is a simple tensor $v \otimes w$?

I know that every vector in a tensor product $V \otimes W$ is a sum of simple tensors $v \otimes w$ with $v \in V$ and $w \in W$. In other words, any $u \in V \otimes W$ can be expressed in the ...
0
votes
1answer
40 views

Exterior power of multilinear functions applied to linearly dependent vectors is zero

I'm working on a homework problem, and we are to show that if $T \in \wedge^p V^*$, and $v_1,\ldots,v_p$ are linearly dependent, then $T(v_1,\ldots,v_p) = 0$. What I've got so far: I understand that ...
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0answers
13 views

Find normal random variables from independent standard normal variables with correlation matrix

I'm trying to find three independent standard normal variables from three normal random variables using a correlation matrix. So far, I have decomposed the problem using the matrix's cholesky ...
19
votes
4answers
450 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
4
votes
2answers
184 views

Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B = ${$v_i : i \in I$} and $W_B = ${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
0
votes
1answer
31 views

What is this map between tensor spaces called? (Change of coordinates).

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, ...