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:
$$\|A^{...
1
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0
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54
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Greatest lower bound and smallest upper bound of Frobenius norm of product of matrices
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}....
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75
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Lower bounding $v^TA^{\dagger}v$ where A is a sum of rank-1 matrices
This is related to a previous question I asked (Upper bounding $v^TAv$ where $A$ is the inverse of a sum of rank-$1$ matrices and $v$ is a vector).
Let $(x_i)_{1 \leq i \leq n}$ be vectors of $\...
5
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2
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256
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Can a norm on polynomials be "almost multiplicative", even for large degrees?
Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
1
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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 ...
1
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0
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345
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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 $p$ ...
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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 $\|x\|+\|y\|$?
11
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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[...
1
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1
answer
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-...
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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
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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 ...
1
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1
answer
859
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A lower bound for the condition number matrix
I have the following proposition:
Theorem: For every invertible matrix $A\in\mathbb{R}^{n\times n}$ and every matrix norm $\|\cdot\|$, then the condition number $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
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1
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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\...
0
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1
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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., $\|...
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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 ...