Questions tagged [operator-norms]
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119
questions
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Does this matrix norm inequality have interesting application in other areas of mathematics?
In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices:
Theorem 3. Let $A=[a_{ij}]$ be a real symmetric ...
0
votes
1
answer
198
views
Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
0
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0
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In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel
For a fixed distribution $p^{in}$ on the input space $ \mathbb{R}^{n_0}$,
consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$.
On this space, ...
12
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0
answers
129
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Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
1
vote
0
answers
44
views
Tail bound on largest singular value of Gaussian Wigner matrix
I have problem on deducing the following tail bound on largest singular value of Gaussian Wigner matrix
$\|W\|\leq(2+\epsilon)\sqrt{n}$, $\forall\epsilon$, with high probability.
There is a hint: see ...
3
votes
2
answers
1k
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Is the matrix induced L1-norm greater than the induced L2-norm?
For $A \in \mathbb R^{m \times n}$ and the induced norms:
$$
\| A \|_1 = \max_{x \ne 0} \frac{\|Ax\|_1}{\|x\|_1}
$$
$$
\| A \|_2 = \max_{x \ne 0} \frac{\|Ax\|_2}{\|x\|_2}
$$
... where:
$$
\|x\|_1 = \...
1
vote
0
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183
views
Schatten norm inequality
Let $A,B$ be two $n\times n$ matrices.
Find a lower bound of the $p$-th Schatten norm
$\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
1
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0
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194
views
How to numerically compute the operator norm of an operator acting on a matrix algebra?
Let $M_n(C)$ denote the $n\times n$ matrices with complex entries acting on the Hilbert space $C^n$. As norm on $M_n(C)$ we take the operator norm, i.e. the largest eigenvalue of its absolute value. ...
0
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0
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Explicit description for dual to operator space of Hilbert space
Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$.
It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....
3
votes
0
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Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix
Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
0
votes
1
answer
251
views
Application of the Frechet derivative [closed]
$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that:
\...
1
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Equivalence constants for induced matrix norms
Disclaimer: I asked this question beforehand on mathematics stack exchange, but I think it is better suited for this site
Given two sets $P_i\in\mathbb{R}^s$, bounded, convex, with non-empty interior ...
5
votes
1
answer
326
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Euclidean norms of matrices
Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the associated operator norm on 2 by 2 matrices.
$$
\left\lVert\pmatrix{a&...
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0
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How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]
Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$.
I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$.
My belief is that this is true is motivated by empirical ...
3
votes
1
answer
487
views
Trace norm of operators obtained by restricting the matrix of a trace class operator
Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...
1
vote
0
answers
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views
Row-wise conjugation of completely bounded map by group action
Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
9
votes
1
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Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...
4
votes
1
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193
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Operator norm of a masked SDP matrix
Let $\Sigma$ be a $d\times d$ semi-definite positive matrix (SDP). Let $I\subset\{1,\ldots, d\}\times \{1, \ldots, d\}$ be a symmetric subset of indices (i.e. if $(p,q)\in I$ then $(q,p)\in I$). We ...
1
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1
answer
173
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Some estimates on tensor norms
Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
2
votes
1
answer
564
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Confusing definition of homogeneous Sobolev norm of order -1
Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...
1
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1
answer
181
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Example when Kantorovich condition would not hold
Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
1
vote
1
answer
130
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Embedding Hermitian forms into Hilbert spaces
Let $H$ be a nondegenerate, not positive definite, Hermitian form on a complex vector space $V$ such that
$$|H(x,y)|\le u(x)u(y)\tag{B}$$
for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ ...
7
votes
1
answer
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Operator norm and spectrum
I am wondering about when an operator norm coincides with the maximum eigenvalue of an operator and there is one particular aspect that confuses me quite a lot.
Let's say we have a symmetric positive ...
9
votes
1
answer
522
views
What is the Lipschitz constant of the differential of the matrix exponential $\mathfrak{so}(3)\to \mathrm{SO}(3)$
I'm interested in numerical methods on $\mathrm{SO}(3)$ manifold, and working on a particular problem using the exponential coordinates:
$$
R(u) := \exp(u_\times)
$$
with $u\in \mathbb{R}^3$ and where ...
6
votes
1
answer
274
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Recover approximate monotonicity of induced norms
Let $A$ some square matrix with real entries.
Take any norm $\|\cdot\|$ consistent with a vector norm.
Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$.
Moreover, ...
3
votes
0
answers
131
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An elementary proof of Davies' inequality
In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality.
Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
3
votes
1
answer
114
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Flatness directions of the operator norm
It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...
2
votes
0
answers
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which operators are "really truly positive"?
Let's say that an operator G on a Hilbert space $\mathcal{H}$ is "really truly positive" iff
$\Vert\exp(-tG) \exp(-tG^*)\Vert_{op}<1$ for all $t>0$
How can we characterize the set of operators ...
0
votes
1
answer
70
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Equality between two norms on $\mathcal{L}(E)^n$
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ the algebra of all bounded linear operators on $E$.
On $\mathcal{L}(E)^n$, we have two equivalent norms:
\begin{eqnarray*}
N_1({\bf A})
&=&...
1
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0
answers
324
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Is the square of operator norm smooth? [closed]
Is $A \mapsto \|A\|_{op}^2$ a smooth function? Specifically, is the space of matrices equipped with operator norm a 2-smooth Banach space?
7
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2
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The norm of tensor product operator on Lp spaces
Let $X, Y$ be two $\sigma$-finite measure spaces and $p,q\in [1,\infty]$. Let $T_1, T_2:L^p(X)\rightarrow L^q(Y)$ be two bounded linear operators. Then one can define a linear operator $$T_1\otimes ...
5
votes
0
answers
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Schur norm of weighted Cauchy matrix
The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value.
Let $a_1,\ldots, ...
0
votes
1
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174
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Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?
Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
3
votes
2
answers
369
views
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
1
vote
1
answer
464
views
Relationship between $2 \to 2$ norm and $\infty \to 2$ norm [closed]
I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is.
E.g., the trivial result is that for matrix $A$ with ...
1
vote
1
answer
601
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Minimal value of matrix norm induced by a norm
Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by
$$
\| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}}
$$
where the matrix $A$ is interpreted as an ...
1
vote
1
answer
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Inequality for the operator norm of a product of matrices
I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that
$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
5
votes
0
answers
153
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Operator norm of a soft thresholded symmetric matrix
Let $A$ be a symmetric real-valued $n\times n$ matrix and let ${\left\|A\right\|_{2\rightarrow 2}} := \max_{\left\|u\right\|_{2}\leq 1} \left\|Au\right\|_{2}$ denotes its operator norm (largest ...
16
votes
4
answers
778
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$\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$
I'm interested in
$$\sup_{x, y} \left\| A x + B y\right\|_2$$ subject to
$$\left\|x\right\|_2 = \left\|y\right\|_2 = 1$$
where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of ...
0
votes
1
answer
244
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Quasiconvexity property of quasinorms
Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.
If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
3
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0
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Araki's proof of simple connectedness of the restricted orthogonal group
I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these ...
0
votes
1
answer
83
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On an error bound for matrix constraints
Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$.
Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...
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votes
5
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Matrix trace & norm [closed]
For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have
$$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$
where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
1
vote
0
answers
75
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specific sequence of matrices making a strange ratio of matrix norms diverging
For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
1
vote
1
answer
120
views
Applying backward shift operator on the composition operators on Hardy spaces
Let $H^2$ be the Hardy space. Let $K_\theta=H^2\ominus\theta H^2$, $\theta$ inner. Let $S$ be the shift operator on $H^2$. Its compression to $K_\theta$ is $S_\theta$.
My question: do we have in the ...
1
vote
1
answer
111
views
Reference request: Riesz operator over Hölder class
We have been told that the Riesz potential in $\mathbb{R}^d$, $I_{\alpha}(f)$, defined by
$$I_{\alpha}(f)(x):= C\int_{\mathbb{R}^d} \frac{f(y)}{\left| x-y \right|^{d-\alpha}}\,\mathrm{d}y $$
has the ...
9
votes
1
answer
494
views
Regular $p$-norm of a matrix
Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
3
votes
2
answers
872
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Determinant of Jacobian and directional derivatives
I have a function $f: \Re^2 \to \Re^2$ and would like to understand why
$$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$
that is, why the determinant of the Jacobian of $f$ at $...
-1
votes
1
answer
115
views
Hierarchies of Operator Norms [closed]
Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e.
$$
\| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
2
votes
1
answer
107
views
Retractions for completely positive unital maps, and their effect on spectral diameter
Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf ...