Questions tagged [tensor-products]
For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.
4,371
questions
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39
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$R$-module structure of tensor product of $R$-algebras (Atiyah-Macdonald)
I am (re-)reading the section on tensor products of $R$-algebras in Atiyah-Macdonald, where $R$ is a commutative ring, and I am not sure about the given definition of the $R$-module structure on the ...
- 2,185
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1
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51
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How is it justified to take the tensor product of nonlinear functions?
Suppose $f, g: \mathbb{R}^n \rightarrow \mathbb{R}^n$. If $f$ and $g$ are both linear then we may define the tensor $$(f \otimes g)(x_1,\ldots,x_{2n}) = f(x_1,\ldots,x_n)g(x_{n+1}, \ldots,x_{2n})$$ by ...
- 4,311
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0
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72
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Can a Riemannian metric be defined in terms of the cotangent space?
I have always thought of Riemannian metrics as being an inner product assigned to each tangent space. That is, if $M$ is a manifold, then at any point $p \in M$,
$$g_p: T_pM \times T_pM \rightarrow \...
- 4,311
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1
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31
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Tensor product in a noncommutative division ring
I have some problem understanding this proof by Qiaochu Yuan
During the proof of the double centralizer theorem, he wrote the tensor product of two module in this way.
Given $T\subset$ End$(A)$ and $T'...
- 105
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0
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33
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Continuity of Linear Map on Tensor Product Spaces with Different Norm Properties
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and extending this to a map $\phi^k: V^{\otimes k} \rightarrow U^{\otimes k}$ defined by
$$
\phi^k(v_1 \otimes \...
- 1,009
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votes
1
answer
21
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operation rules involving tensor product of perturbed wavefunctions
Let $i, j$ represent a wavefunction.
In Bra - Ket notation, I have an expression like
$\langle ij | g | i j \rangle $ which is also just $(\langle i | \otimes \langle j | ) | g | (|i \rangle \otimes |...
- 6,081
0
votes
1
answer
49
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The categories and functors implied on this definition of tensor product [duplicate]
In wikipedia the definition of tensor product by the universal property can be found here. It says that:
The tensor product of two vector spaces V and W is a vector space denoted as V ⊗ W, together ...
- 954
1
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1
answer
66
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Why is $\pi(\sigma)$ a well defined mapping $V^{\otimes n} \to V^{\otimes n}$?
Given an $R$-module V, where $R$ is a commutative ring (we even assume $\mathbb{Q}$ is a subring, though I don't think that's required for my particular question), we wish to view the wedge product $V^...
- 163
1
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1
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48
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Universal Property of tensor product of $\mathbb Z_2$-graded algebras.
If $A$ and $B$ are two $k$ algebra's with a $\mathbb{Z}_2$-grading then I know that a $\mathbb{Z}_2$-graded structure can be defined on their tensor product. One does this by altering the ...
- 13
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0
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34
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Understanding the order of tensor application as a linear map.
I have a question regarding the application of one tensor to another, $Q(D)$. Let's start with the simplest example, considering the bilinear form $G=G_{ij}(\epsilon^{i}\otimes \epsilon^{j})$ acting ...
- 13
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0
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10
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Exactness and tensor product questions based on $A/\mathfrak{a} \otimes M \cong M/\mathfrak{a} M$
My brain has refused to understand tensor products for a very long time, so I wrote up a solution for this classical exercise accompanied by my elementary questions.
Exercise: Let $A$ be a ring, $\...
1
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0
answers
31
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Misunderstanding of the definition of the tensor product of algebras.
In Atiyah it says that if B,C are A-algebras with $$ f:A\to B~ ~and~~ g:A\to C$$ then their tensor product is D and a ring homomorphism $$h:A\to D~~~ with ~~a\mapsto f(a)\otimes g(a)$$ I don't ...
- 31
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1
answer
66
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Are irreducible elements of the tensor product of a vector space equivalent to irreducible polynomials?
I'm looking for some feedback on a construction I came across. Loosely, it entails sending a vector space isomorphically to a vector space of polynomials, 'restoring' the ring structure, and asking ...
- 83
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60
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What is the difference between the tensor product of a covector and a vector and a covector acting on a vector as a 1-form?
I believe I'm getting quite confused between the tensor product and the dual vector as a 1-form acting on a usual vector. Essentially, I'm struggling to distinguish the difference between $\epsilon^{i}...
- 13
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1
answer
53
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If $M$ and $N$ are finitely generated $R$-modules such that their tensor product is $0$, then their annihilators over $R$ are comaximal
I am trying to prove this using the local-global principle. Let $I = Ann_R(M) + Ann_R(N)$.
It suffices to show that $I_\mathfrak{m} = R_\mathfrak{m}$ for all maximal ideals $\mathfrak{m}$ containing $...
2
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1
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51
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Does the Kronecker product for matrices satisfies the universal property of the tensor product (of modules)?
The Kronecker Product is defined here: https://en.m.wikipedia.org/wiki/Kronecker_product
It is sometimes also called 'tensor product'.
Hence, I would like to know whether it satisfies the universal ...
- 1,292
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0
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41
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Understand the tensor product over a field
I often use the tensor product but I don't understand it very well despite that I know his universal property and an example of his construction, for me it is like a black box and I would like to ...
- 118
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1
answer
66
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Decompositions of exterior products
Let $V$ be a real vector space equipped with an almost complex structure $J$ and denote the complexification by $V_\mathbb C$. The complexified vector space splits into the eigenspaces $V^{1,0}$ and $...
2
votes
1
answer
55
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Projective tensor product of operator spaces
Consider the following fragment from Effros and Ruan's book "Operator spaces"
Why is a decomposition as in the red box possible? In fact, it is not even clear to me that any such ...
- 356
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0
answers
27
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Computer algebra system for Tensor Algebra
I am searching for a computer algebra system that allows to tackle the following problem:
Let $V = \mathbb{R}^3$. Consider the free commutative algebra $\mathrm{S}^\bullet V$, with $\mathrm{S}^kV$ ...
- 163
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0
answers
21
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Isomorphisms of the tensor product of $\mathcal{A}^N$ [duplicate]
Let $\mathcal{A}$ be an algebra and define $\mathcal{A}^N:=\mathcal{A}\oplus\dots\oplus\mathcal{A}$.
I need to show that
$\mathcal{A}^N\otimes_\mathcal{A} \mathcal{A}^N\cong M_N(\mathcal{A})$
$\...
2
votes
2
answers
65
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Representations of direct products of groups
Each irreducible representation of $G_1 \times G_2$ is isomorphic to a
representation $\rho^1 \otimes \rho^2$, where $\rho^i$ is an irreducible representation of $G_i$ ($i = 1,2$).
Can we extend this ...
- 1,953
2
votes
1
answer
78
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Are there other "expressions" of $a \otimes b$ that are isomorphic to $V \wedge V := \textrm{span}\{a \otimes b - b \otimes a\}$? [duplicate]
We usually define $V \wedge V := \textrm{span}\{a \otimes b - b \otimes a; \ a, b \in V\}$. If $V$ is a vector space over $\mathbb{K}$, it's obvious that $k(V \wedge V) \cong V \wedge V$ for any ...
- 616
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0
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29
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Are there Plucker-like relations for the tensor product of two decomposable differential forms?
Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form
$$
\mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
- 93
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0
answers
28
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Doubt concerning the definition of the tensor product of modules
Let $R$ be a ring with unit and consider a right $R$-module $M$ and a left $R$-module $N$. The tensor product $(M \otimes_R N, \otimes_R )$ of $M$ and $N$, is usually defined as the quotient of the ...
- 572
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$M$ $\mathscr{R}$-submodule complete lattice $\iff$ $aL \subset M \subset a^{-1}L$, $a \in \mathscr{R}$
My question may be nestled in the sense that there might be confusion about several notions, please let me know if it is the case.
In "The Arithmetic of Hyperbolic 3-manifolds" by Maclachlan ...
- 824
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1
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52
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Change of basis in tensor notation
When trying to deeply visualize the meaning of the entries of a basis transformation matrix, I realized that a (forward) change of basis matrix can be written in a tensor form like this:
$$Q=Q^i_j \; \...
- 3,984
1
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0
answers
25
views
Meaning of injective tensor product
Let $X, Y$ be two locally convex topological vector spaces. I can tell myself a story to make $X\otimes_{\pi} Y$ and $X\otimes_{\iota} Y$ (the projective and inductive tensor products, respectively) ...
- 300
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1
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Are two linear mappings $M$ and $N$ on $\mathcal{S}(H_A \otimes H_B)$ equal if they have same behaviour on product states? [closed]
Let $M$ and $N$ be mappings from $\mathcal{S}(H_A \otimes H_B)$ to itself, where $\mathcal{S}(H)$ denotes the set of density operators over the Hilbert space $H$.
If the following two conditions hold:
...
- 55
3
votes
0
answers
56
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Tensor vs elementwise product notation
I am familiar with undergrad level linear algebra, but we have not covered neither tensors nor anything more algebraic than a matrix/dot product. I do perfectly understand what an elementwise product ...
- 35.5k
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2
answers
84
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Prove naturality of unit and counit in Tensor-hom adjunction
I am learning some basics from Category Theory and I am currently working on understanding Tensor-hom adjunction. I was reading a Wiki article about it (https://en.wikipedia.org/wiki/Tensor-...
- 587
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0
answers
53
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Prove that $\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] = \sum_{k} E_{k}\rho E_{k}^{\dagger}$
In section 8.2.3 of Nielsen and Chuang, there is a derivation of the operator-sum representation as follows:
$\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] \tag{1}$
And $\...
- 21
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vote
1
answer
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trace of wedge product and cyclic property [closed]
Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein.
If I am taking the trace of a wedge product of matrices,...
- 19
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votes
1
answer
44
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Traceless tensor product: unclear definition
I do not understand here on the page $8$
that in the definition of traceless tensor product there are 3 indices $i,j,k$
in the rightmost term but in the preceding ...
- 3,802
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votes
0
answers
33
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R is torsion free is if and only if the natural map is injective
I'm reading Silverman's Arithmetic of Elliptic Curves which has the following statement:
A ring R is torsion free is equivalent to saying, the map R$\longrightarrow$R$\otimes\mathbb{Q}$ is injective.
...
- 181
1
vote
1
answer
64
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Prove Tensor-hom adjunction via Hom-set definition
I am trying to learn some basics of category theory, precisely Adjunction, but I've encountered some difficulties trying to prove such a statement.
So, I want to prove that tensor product $- \otimes X$...
- 587
4
votes
1
answer
54
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Associativity of tensor product for (bi)modules
Theorem IV.$5.8$ of Hungerford's Algebra:
If $R$ and $S$ are rings and $A_R$, ${}_{R}B_{S}$, and ${}_{S}C$ are (bi)modules, then there is an isomorphism $$(A \otimes_RB)\otimes_S C \cong A \otimes_R(B\...
- 1,953
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2
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166
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Eigenvalues of Ad$X$ are $\lambda_j-\lambda_k$ where $\lambda_i$'s are those of $X$
Prove the following proposition.
If square matrix $X$ has $n$ eigenvalues $\{\lambda_j|j=1,2,\dots,n\}$, then $\operatorname{ad}X (\operatorname{ad}X(Y):=[X,Y]:=XY-YX,\,\forall$ square matrix $Y$ of ...
- 9,583
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1
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59
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Meaning of computing the tensor product of vectors
In a nutshell: a proposition in Abraham's Foundations of Mechanics states that, given a vector space $E$ with basis $e_1,\ldots,e_n$, the space $T^r_s(E)$ of tensors of type $(r,s)$, the dual basis $\...
- 4,708
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0
answers
40
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Exact sequence of tensor products when a module homomorphism is not an epimorphism
The following theorem is famous in module theory and is Proposition IV.$4.3$ of Hungerford's Algebra.
I) $A \overset{\theta}{\rightarrow} B \overset{\xi}{\rightarrow} C\rightarrow 0$ is an exact ...
- 1,953
2
votes
1
answer
84
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Does $\operatorname{Ann}_B(B \otimes_A M) = (\operatorname{Ann}_A(M))^e$ when $M$ is finitely generated? [duplicate]
I am working on solving part (viii) of exercise 19 in chapter 3 of Atiyah-MacDonald. In the problem, we have rings $A, B$, a ring homomorphism $f : A\to B$, and a finitely generated $A$-module $M$; we ...
- 1,153
3
votes
1
answer
60
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Tensor product of matrix modules over matrix ring
For positive integers $m$ and $n$ and a field $k$, write $\mathbf{M}_{m\times n}(k)$ for the $k$-vector space of $m\times n$ matrices with entries in $k$. In particular $\mathbf{M}_n(k):=\mathbf{M}_{n\...
- 2,402
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0
answers
15
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Book/articles about computing the high-order derivative of a vector field
Is there any book/article that gives a general result of this:
For any $n\in \mathbb{N}^*$, use the chain rule to compute the $n$-th order derivative of :$(f_1(x_1(t),x_2(t),\dots,x_n(t)),f_2(x_1(t),...
- 11
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1
answer
83
views
How is the arithmetic multiplication related to tensor products?
Mathematical objects are supposed to "model things", sometimes, more mathematics. In this sense, i think is "natural" to assume that tensors, being such universal objects as they ...
0
votes
0
answers
13
views
Explanation of a a differential of a tensor with respect to its FFT.
I found in a paper that, for a tensor $\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$
$$\nabla || \mathcal{X}||= \frac{\partial ||\mathcal{X} ||}{\partial \overline{\mathcal{X}}} \times_3 F^{*...
0
votes
2
answers
52
views
Rank and dimension dilemma for modules
Let $R$ be a field of characteristic zero, and $M$ be a $\mathbb{Z}$-module. Knowing that $R\otimes M$ is a vector space, can we deduce that $M$ is a free $\mathbb{Z}$-module and $\text{rank}(M)=\...
- 1,785
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0
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26
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Does bases of operators on two Hilbert spaces span the operators on the tensor product of the two spaces?
Let $\mathcal H_1, \mathcal H_1$ be two Hilbert spaces, $\mathcal H=\mathcal H_1\otimes\mathcal H_2$ is their tensor product. Let $\mathcal L(\mathcal H_1),\mathcal L(\mathcal H_2),\mathcal L(\mathcal ...
- 133
1
vote
0
answers
45
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Tensor Product of Vector with Itself?
Let $\vec x\in \mathbb{R}^n$, and $A\in\mathbb{R}^{m\times n}$.
If I wanted to compute a linear map
$$\vec x\mapsto A\vec x,$$
it would suffice to compute it on $n$ unit vectors $u_i = A\vec e_i$.
...
- 6,918
2
votes
2
answers
73
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Compute the tensor product $\mathbb{Z}\otimes_{\mathbb{Z}[x]}\mathbb{Z}$
Question:
For each $n\in\mathbb{Z}$, define the ring homomorphism
$$\phi_n:\mathbb{Z}[x]\to\mathbb{Z}$$
by $\phi_n(f)=f(n)$. This gives a $\mathbb{Z}[x]$-module structure on $\mathbb{Z}$, i.e,
$$f\...
- 499
0
votes
0
answers
51
views
Properties of unitary transformations of random matrices
I'm trying to find an upper bound for the following expression
$$
\mathbb{E} || \Psi_2^{+} \Psi_1 A ||_F
$$
Where $A \in \mathbb{R}^{(n-k) \times m}$ is fixed, and the matrices $\Psi_1 \in \mathbb{R}^{...
- 951