All Questions
Tagged with upper-lower-bounds normed-spaces
60
questions
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The lower bound of Frobenius norm of matrices product.
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, where $\|.\|$ is Frobenius norm.
I would like to find the lower bound of $\|A^{-1}.B.(A+B)^{-1}\|$:
I have the result:
$$\|A^{...
- 11
1
vote
0
answers
54
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Greatest lower bound and smallest upper bound of Frobenius norm of product of matrices
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}....
- 11
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1
answer
75
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Lower bounding $v^TA^{\dagger}v$ where A is a sum of rank-1 matrices
This is related to a previous question I asked (Upper bounding $v^TAv$ where $A$ is the inverse of a sum of rank-$1$ matrices and $v$ is a vector).
Let $(x_i)_{1 \leq i \leq n}$ be vectors of $\...
- 192
1
vote
1
answer
173
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Lower bound of norm squared
Let $(X, \| \cdot \|)$ be an $n$-dimensional normed
$\mathbb{R}$-linear space. Let $\{x_1, x_2, ..., x_n\}$ be a basis of $X$. Show
that there exist positive constants $C_1$ and $C_2$ such that for ...
- 580
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0
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344
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Bound on L2-Norm of Probability Distributions?
I am working on a problem where I have continuous probability distributions $p$ over a bounded domain $D$, i.e., $\forall x$ $p(x) \geq 0$ and $\|p\|_1 = \int_D p(x) dx = 1$. However, I also want $p$ ...
- 127
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23
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How to find an upper bound for a norm with power less than 1?
I need to find an upper bound for $$\|x-y\|_2^\nu$$ in which $\nu \in [0,1]$. I was wondering if $\|x-y\|_2^2$ can be considered as an upper bound. or $\|x\|+\|y\|$?
1
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1
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79
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If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If is not true in general, please give some counter-...
- 1,558
11
votes
1
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325
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Can a norm on polynomials be supermultiplicative?
A norm on a real algebra is supermultiplicative when $\lVert f\cdot g\rVert\geq\lVert f\rVert\cdot\lVert g\rVert$ for all $f$ and $g$ in the algebra.
Is there a supermultiplicative norm on $\mathbb R[...
- 5,726
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2
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256
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Can a norm on polynomials be "almost multiplicative", even for large degrees?
Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
- 5,726
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0
answers
485
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Upper bound on the norm of an inverse of a positive definite matrix
Given two real-valued symmetric positive-definite matrices $A$ and $B$. Assume that $A\succeq I$. Let $\Delta = B-A$. We are interested in a bound on $\Vert B^{-1}-A^{-1}\Vert$ in terms of $\Vert \...
1
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1
answer
422
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Upper Bound for the Induced 2-Norm $\|(A+B)^{-1}A\|$
Assume $A$ is a positive definite matrix, and $B$ is a positive semi-definite matrix. I am interested in the problem of whether there exists a constant upper bound for the induced 2-norm (spectral ...
- 163
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votes
1
answer
201
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supremum on quadratic and bilinear form
Let $V$ be a real normed vector space. Denote by $V_1$ the subset of $V$ of vectors with norm $\le 1$. Consider a positive-semidefinite symmetric bilinear form $\langle \cdot, \cdot\rangle : V\times V\...
- 8,127
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1
answer
335
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bounds on norm of a vector
Let $x \in \mathbb{R}^n$, and $M \in \mathbb{R}^{n\times n}$ be a full rank square matrix. If it is known that $ \| M x\|_2 \leq c$, then what can be said about the upper bound of $\| x\|_2$, i.e., $\|...
- 1,722
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1
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72
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Lower bound on the components of a unit vector
Let $u \in \mathbb{R}^n$ be a unit vector: $\|u\| = 1$. Is there a formula for the smallest $C > 0$ such that $|u_i| \geq C$ for some component $u_i$ where $i = 1,...,n$?
For example, if $u \in \...
- 1,093
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0
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95
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If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?
Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$?
My Approach:
$$\...
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votes
1
answer
90
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Randomness in the norm of sum of vectors
Let $x_1,x_2,\ldots,x_n \in \mathbb{R}^d$ be vectors and $a_1, a_2, \ldots, a_n \in \mathbb{R}$ be random iid scalars distributed by $N~(0,\sigma^2).$
Then, is it possible to lower bound the ...
-1
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1
answer
55
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$\ell_2$ vs $\ell_{\infty}$ induced norm of a square matrix
I'm wondering if the result of this post $\| A \|_{L^2} \le \| A \|_{\infty}$ for symmetric matrices $A$ applies to square matrices that are not symmetric. Of course, for an asymmetrical square ...
- 155
1
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0
answers
76
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Probability of ell-1 norms of vertices of the rotated Hamming cube
Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
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84
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Gettings bounds for seminorms from bound of absolute value
On a compact domain $\Omega \subseteq \mathbb{R}^d$, we have a function $u(x) \in C^{\infty}(\Omega)$ with an approximation $u_h(x)$ with the following properties:
$$
|u(x) - u_h (x)| \leq C h^{m+1} |...
- 115
1
vote
1
answer
36
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Wrong defined/boundless $g \rightarrow\sum_{n = 1}^\infty\frac{g(\frac1n)}{2^n}$
Let's consider:
$$f\colon (C[0,1], \Vert\cdot \Vert_1) \ni g\rightarrow \sum_{n=1}^\infty\frac{g(\frac1n)}{2^n}$$
I'm trying to check if this object is well defined.
where $\Vert f \Vert_1 = \int_0^1|...
- 1,787
3
votes
3
answers
363
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Why $\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)}$
Given $A\succ 0$ (positive-definite) and $c>0$, I am trying to show
$$\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)} \tag{1}$$
using information like this but without success so far. Could ...
- 865
1
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1
answer
51
views
Inconclusive inequality result: show $(id - f)^{-1}$ is bounded
I have the following statement to prove:
Let $V$ be a normed vector space (so not necessarily complete or finite dimensional) over $\mathbb{R}$. Take any norm $\Vert \cdot \Vert_V$ on $V$ and let $\...
- 155
1
vote
2
answers
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Bound on l1 norm given bound on l2 norm
While doing self-study exercices, I found the following bound without explanation and was not able to see why it is always the case. I found some examples, it seems legitimate but I am unable to ...
- 91
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1
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Is the 2-norm of a matrix bounded by the maximum of its 1-norm and Infinity-norm?
I am implementing the algorithm in "Approximating the Logarithm of a Matrix to Specified Accuracy" by Sheung Hun Cheng, Nicholas J. Higham, Charles S. Kenny, Alan J. Laub, 2001.
In this ...
- 153
1
vote
1
answer
130
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Bounding $\|A^n-B^n\|_F$ by $\|A-B\|_F$
Given
$$ \epsilon = \|A-B\|_F, $$
it is clear that
$$ \epsilon^n > \|(A-B)^n\|_F, $$
which follows from submultiplicativity.
I wonder if something related can be said about $\|A^n-B^n\|_F$?
- 195
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1
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53
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Bound for $\Vert\sum_iA_i^\dagger A_i\Vert_\infty$ given that $\Vert\sum_i A_i\Vert_\infty$ is small
Let $A_i$ be matrices such that
$$\left\Vert \sum_i A_i \right\Vert_\infty \leq \varepsilon,$$
where $\Vert\cdot\Vert_\infty$ is the operator norm and is equal to the largest singular value of its ...
- 1,708
3
votes
2
answers
400
views
How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
- 81
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votes
1
answer
51
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alternative asymptotic bounds
I have an $n$ by 1 vector of weights $w$, and an $n$ by $k$ matrix, $\Gamma$. I have that $w'w$ is $\mathcal{O}(1)$, $\frac{\Gamma'\Gamma}{n}=\mathcal{O}(1)$ and $\frac{\Gamma\Gamma'}{n}=\mathcal{O}(1)...
- 37
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1
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28
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Deducing properties of the $\ell_3$ norm from the $\ell_1$ and $\ell_2$ norms
Suppose we have a function $f: [0,1] \rightarrow \mathbb{R}$, $f(x) \geq 0$ normalised so that $\|f\|_1 = 1$, where
$$
\| f\|_p = \left( \int_0^1 f(x)^p d x \right)^{1/p}.
$$
Moreover, we know that $\|...
- 1,149
2
votes
0
answers
83
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Upper bound in bayesian regression setting
Let $y_i = x_i^\top \beta + \epsilon_i$, $i=1,\ldots,n$; where $\epsilon_i$ are i.i.d. following a distribution with mean zero and unit variance, i.e., $\epsilon_i \sim P_{\epsilon_i}(0,1)$, $i=1,\...
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