All Questions
Tagged with upper-lower-bounds normed-spaces
60
questions
1
vote
1
answer
93
views
Matrix norms translating with identity matrix
Let $\mathcal{A}$ denote a linear operator, $\mathcal{A}^*$ its adjoint, and $\text{I}$ the identity matrix. Is the norm of $\mathcal{A}$ related to the norm of $\mathcal{A}^* \mathcal{A} - \text{I}$?
...
- 13
0
votes
0
answers
381
views
upper bound for quadratic form in terms of vector norm and eigenvalues
I have a quadratic form. if Q, P and M are positive and symmetric matrices.
$$(-x^T Q x - 2 x^T Q e - e^T Q e) + (y^T M y + 2 x^T P y + 2 e^T P y )$$
how can I get an upper bound for this quadratic ...
- 1
0
votes
1
answer
368
views
Relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$
Suppose we have two matrices $A\in \mathbb{R}^{m \times n}$ and $B\in \mathbb{R}^{n \times p}$, then what's the relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$? The notation $\|\cdot\|_*$ means ...
- 20
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
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 $A$:
$$\|A\|_\infty = \|UDV\|...
- 4,945
1
vote
0
answers
78
views
Trace and 2-norm of linear combination of outer products
Suppose that $c_i \in \mathbb{R}-\{0\}, B_i \in \mathbb{R}^{k \times m}, \alpha \in \mathbb{R}^k$ with $\|\alpha\|_2 = 1$. Consider the following linear combination of outer products:
$$M = \sum_{i=1}...
- 1,180
0
votes
1
answer
35
views
What conditions should vector $x$ satisfy so that $\|[x_2+\alpha x_1, \dots, x_n + \alpha x_1]\|_2$ is bounded by a constant?
Suppose that $x = [x_1,\dots,x_n]$ is a vector with norm less than or equal to one $\|x\|_2^2 \leq 1$. Let $\alpha \in [0,1]$ and define the following vector
$$y = [x_2+\alpha x_1, \dots, x_n + \...
- 1,180
1
vote
1
answer
609
views
Find an upper bound on the expectation of squared norm given an upper bound on the expectation of norm
For non-independent random vectors $X, Y$, I have an upper bound on the expectations $E[\|X\|_2] \leq a, E[\|Y\|^2_2] \leq b$. How can I compute an upper bound for $E[\|X^\intercal Y\|_2]$ or $E[\|X^\...
- 1,180
1
vote
0
answers
60
views
2-norm of product of multiple matrices
Suppose that $A \in \mathbb{R}^{m,n}$ and $B \in \mathbb{R}^{n,m}$ and $A$ has a bounded 2-norm. We know that $AB$ is a positive semi-definite matrix with $\|AB\|_2 \leq 1$. Further, assume that $C \...
- 1,180
0
votes
1
answer
122
views
upper bound for the norm of vector belonging to the column space of a matrix
If $u$ is a vector that belong to the column space of a matrix $A$ and $\sigma_{\min}(A)$ is the smallest non zero eigenvalue of $A^T A$, then I read in a paper that we can write
$$ \sigma_{\min}(A)^{\...
- 105
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
0
votes
1
answer
21
views
$L_1$ distance between re-normalized points
Let $\mathbf{x},\mathbf{y} \in (0, \infty)^d$. Are there general relations between
$
\Vert \mathbf{x}-\mathbf{y} \Vert_1
$
and
$
\left\Vert
\frac{\mathbf{x}}{\Vert \mathbf{x}\Vert_1}
-\frac{\mathbf{y}...
- 1,786
1
vote
1
answer
133
views
Finding an upper bound on composite $C^{1}$ functions
The conditions for the problem I am currently tackling is as follows:
Let $T, M > 0$. Define :
$X_{T} := \{ u \in C^{1} ([0,T]) : ||u||_{T} \leq 2M \}$
where $||u||_{T} \ = \ ^{\text{sup}}_{t \in (...
- 467
1
vote
0
answers
99
views
Norm equivalence constants
Take a polynomial $g\in\mathbb{R}[\mathbf{x}]$, in $n$ variables and having some degree $d$, with $g(\mathbf{x})\geq 0$. We define the $p$-norms of $g$ as
$$
\vert \vert g \vert \vert _{p} = \left( \...
- 177
1
vote
1
answer
298
views
Bounding the dot product of two planar unit vectors.
Does there exist a continuous, monotone increasing function $f\colon[0,2]\to [0,1]$, satisfying $f(0)=0$ and $f(1)=1$, such that for all vectors $(a_1,b_1),(a_2,b_2)\in \mathbb{R}^2$ of unit length, i....
- 2,684