The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
162 views

Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello, Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero. As a part ...
1
vote
3answers
1k views

Comparison of the L_p norm of a matrix and its entry-wise absolute value

Suppose $A_{n \times n}$ is a matrix and $A' = (|A_{ij}|)$ is its entry wise absolute form, can be give an upper bound and lower bound of the L_p norm $\|A\|_p$ using the L_p norm of the absolute ...
8
votes
2answers
336 views

Projections onto $n$-codimensional subspaces of a Banach space: norms.

Hello, I'd like some help to find an answer I've been looking for since this morning. Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...
3
votes
1answer
1k views

norm of (sub)stochastic matrix

Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.
1
vote
0answers
99 views

expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$. Let $x \in {\...
2
votes
2answers
730 views

How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$...
1
vote
0answers
179 views

What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector? By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$. P.S. This is ...
2
votes
2answers
2k views

Dual Norm For Sum of 2-Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$ $\|\mathbf{x}\| = ...
1
vote
2answers
317 views

An inequality with $\ell_p$ norm

I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the question at math.stackexchange which did not attract much ...
3
votes
3answers
2k views

upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
17
votes
1answer
657 views

Extremal points of the unit ball of M_n(R) …

The unit ball of ${\bf M}_n(\mathbb R)$ is a compact convex subset. As such, it is (Krein-Milman theorem) the convex envelop of its extremal points. So far, so good; but the unit ball depends of the ...
8
votes
0answers
501 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
14
votes
3answers
599 views

Bounding the commutator [A,B] in terms of the numerical radius

Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$ The answer is ...
0
votes
0answers
360 views

Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension

Bonjour/bonsoir à toutes et à tous. This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...
3
votes
2answers
442 views

“Measuring” how far is one Banach space from being surjectively isometric to another

Bonjour/bonsoir à toutes et à tous. Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field). Question 1. What ...
0
votes
1answer
170 views

Truncated Dirichlet series take their supremum on the imaginary axis

Hi there, I am struggling with a theorem about truncated Dirichlet series. I am trying to prove the following theorem: Let $(a_n)_n \subset \mathbb{C}$ and $N \in \mathbb{N}$. Then $\sup_{t \in \...
3
votes
3answers
952 views

Minimum norm of convex hull

Dear all, I am currently stuck at a problem which seems too easy to be stuck at to me... Summary Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute \[\...
4
votes
1answer
1k views

When do 0-preserving isometries have to be linear?

Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$. Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ . What ...
3
votes
0answers
357 views

Bounding linear combination of exponentially decaying functions.

Let $f(x)=e^{-a|x|}$ and $a>0$ a constant. Suppose we take a linear combination $s(x) = \sum_{i=1}^n \alpha_i f(x-x_i)$ where $\alpha_i\in\mathbb{R}$ and $-r< x_1< \ldots < x_n< r$. ...
3
votes
1answer
794 views

Inequalities and bounds for relating p-norms (Reference request)

Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the ...
2
votes
2answers
293 views

Analogue of an orthogonal subspace in a noneuclidian normed space

This question is related to http://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question. We ...
2
votes
0answers
584 views

What is the name for $(a^2 + b^2 + c^2 +…)/(a + b + c +…)$? [closed]

That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.