All Questions
Tagged with upper-lower-bounds normed-spaces
60
questions
11
votes
1
answer
325
views
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
6
votes
1
answer
6k
views
Matrix Norm Bounds
A natural consideration of matrix norms is to compare them, and one of the many standard results on the induced 1, 2, and $\infty$-norms indicate that
$$\frac{1}{\sqrt{n}}\|A\|_\infty\leq \|A\|_2\leq \...
- 1,793
5
votes
2
answers
256
views
Can a norm on polynomials be "almost multiplicative", even for large degrees?
Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
- 5,726
5
votes
0
answers
84
views
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:
$$
|u(x) - u_h (x)| \leq C h^{m+1} |...
- 115
4
votes
1
answer
59
views
After applying a sequence of involutory real matrices to a vector, is the norm of this vector bounded from below?
For $n,N \in \mathbb{N}$, let $A_1, \ldots, A_n$ be a finite sequence of involutory $(N \times N)$-matrices over $\mathbb{R}$, i.e. $A = A^{-1}$.
We know, that the eigenvalues of any involutory matrix ...
- 185
4
votes
0
answers
379
views
Need to improve upper bound for $\| (uv^T + B)^{-1} \|$ (Sherman-Morrison formula)
I have a matrix $A \in \mathbb{C}^{n \times n}$ in the form $A = uv^T + B$, where $u,v \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$. I know $A$ is invertible and want to find an upper bound ...
- 6,574
3
votes
3
answers
363
views
Why $\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)}$
Given $A\succ 0$ (positive-definite) and $c>0$, I am trying to show
$$\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)} \tag{1}$$
using information like this but without success so far. Could ...
- 865
3
votes
2
answers
400
views
How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
- 81
3
votes
0
answers
244
views
Upper bound the maximum column sum of a particular stochastic matrix
Let $(x_i)_{i=1}^N$ be a set of vectors in $\mathbb{R}^D$. Define the matrix $W \in \mathbb{R}^{N \times N}$ as:
$W_{ij} = \frac{\exp(-||x_i-x_j||^2)}{\sum_k \exp(-||x_i-x_k||^2)}$
i.e. row $i$ of $...
- 51
2
votes
2
answers
673
views
If the expectation of the norm of random variable is bounded, does that imply that the expectation of the squared norm is bounded?
Let $\mathbf{x}$ be a random vector. If $\mathbb{E}[\|\mathbf{x}\|]$ is upper bounded, does that imply that $\mathbb{E}[\|\mathbf{x}\|^2]$ is also upper bounded?
The Jensen's inequality goes the ...
2
votes
1
answer
51
views
alternative asymptotic bounds
I have an $n$ by 1 vector of weights $w$, and an $n$ by $k$ matrix, $\Gamma$. I have that $w'w$ is $\mathcal{O}(1)$, $\frac{\Gamma'\Gamma}{n}=\mathcal{O}(1)$ and $\frac{\Gamma\Gamma'}{n}=\mathcal{O}(1)...
- 37
2
votes
1
answer
98
views
Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $. [closed]
I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it.
Show that, for each $...
- 251
2
votes
0
answers
83
views
Upper bound in bayesian regression setting
Let $y_i = x_i^\top \beta + \epsilon_i$, $i=1,\ldots,n$; where $\epsilon_i$ are i.i.d. following a distribution with mean zero and unit variance, i.e., $\epsilon_i \sim P_{\epsilon_i}(0,1)$, $i=1,\...
- 81
2
votes
1
answer
227
views
Show that $\|A\|_{\infty} \leq \sqrt n \|A\|_2$
Question:
Let $A \in \Bbb R^{m\times n}$. Show that $$\|A\|_{\infty} \leq \sqrt n \|A\|_2$$
Attempt:
First, I tried invoking the SVD (Singular Value Decomposition) of $A$:
$$\|A\|_\infty = \|UDV\|...
- 4,945
1
vote
2
answers
3k
views
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