Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
9
votes
3answers
604 views
How many elements in a number field of a given norm?
Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
4
votes
1answer
192 views
$L_1$ projection of sum of convex functions onto polytopes
Suppose I have a function $f(x) : \mathbb R^n \to \mathbb R$ that is the sum of a given strictly convex function $g : \mathbb R \to \mathbb R$ in a single variable, i.e. $f(x) = g(x_1) + g(x_2) + ...
2
votes
1answer
400 views
Multiplication operator norm
I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem,
For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
1
vote
1answer
81 views
What is the proper term for the entity that relates a vector space and a set?
One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
5
votes
1answer
107 views
a conjecture on norms and convex functions over polytopes
Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
2
votes
1answer
124 views
Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$
Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence
...
2
votes
0answers
100 views
Quick question: finite extensions and norms
[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."]
I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
8
votes
2answers
122 views
Finding all scalars $k$ such that $\| kv \| = 10$
I have a homework question that asking to
Find all scalars $k$ such that $\|kv\| = 10$ when $v=(1,-4,6)$.
What I did that that I found the norm of $v$ which I found to be $\sqrt{53}$. Then I ...
3
votes
1answer
369 views
A question on linear transformation
Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard
matrix is
$$\left(\begin{array}{rrrr}
1 & -1 & -1 & -1\\
1 & 1 & 1 & -1\\
1 & 1 & -1 & 1\\
...
2
votes
1answer
96 views
Example of a vector norm for which $\|I\|<1$
In order to prove a larger assumption, I need to find a vector norm over $M_n$ such that $\|I\| < 1$. None of the standard $p$-norms, nor the infinity norm work. I know that for matrix norms, ...
1
vote
1answer
47 views
Equivalence of a Vector Norm being Absolute
I'm trying to show that a vector norm $\|\cdot\|$ being absolute ($\|x\| = \|\;|x|\;\|)$ is equivalent to showing that $\|x'\| = \|[\alpha_1x_1\ldots\alpha_nx_n]^T\| = \|x\|$ for all ...
2
votes
1answer
226 views
Matrix Norm Identity Derivation
I am having trouble figuring out where something in a book I am reading is coming from. (The book is Matrix Computations by Golub and Van Loan, 3rd edition, p.58.) It will probably be obvious once ...
1
vote
1answer
274 views
Cauchy sequence in a normed space
Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent.
I suspect the following to be true.
Let $(x_n)_{n=0}^\infty$ ...
2
votes
3answers
107 views
Convergence of a pair linearly independent elements of a vector space
Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
1
vote
0answers
97 views
Help with a proof that the inversion mapping for matrices is continuous
I'm reading a proof that the map $A\mapsto A^{-1}$ is continuous in the operator norm. A part of the proof is that if $A,B$ are matrices such that $A$ is invertible and $\beta=\parallel ...
2
votes
1answer
420 views
Subordinate matrix norm
I have the following matrix norm:
$$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$
I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
2
votes
2answers
1k views
how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?
In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof:
$$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$
...
1
vote
1answer
126 views
Maximum 1-norm of matrix times unit vector
If $A$ is a $p \times p$ matrix, what is
$$\max_{||u||_2=1} ||Au||_1 ?$$
I am specifically interested in the case when $A$ is positive definite.
3
votes
1answer
650 views
Is there a geometric meaning of the Frobenius norm?
I have a positive definite matrix $A$. I am going to choose its Frobenius norm $\|A\|_F^2$ as a cost function and then minimize $\|A\|_F^2$. But I think I need to find a reason to convince people it ...
3
votes
2answers
96 views
Geometric Sums in Banach Algebra
Let $E$ be a Banach algebra, and $v\in E$, so that $||v|| < 1$. So the geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm, that part I understand.
I can show that $||w|| \le ...
4
votes
1answer
183 views
p-adic norms and products
I came across the following problems about p-adic norms:
Problem. Show that $$\prod_{p} |x|_p = \frac{1}{|x|}$$ where the product is taken over all primes $p = 2,3,5, \dots$ and $x \in ...
3
votes
1answer
71 views
Does Permuting the Rows of a Matrix $A$ Change the Absolute Row Sum of $A^{-1}$?
For $A = (a_{ij})$ an $n \times n$ matrix, the absolute row sum of $A$ is
$$
\|A\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |a_{ij}|.
$$
Let $A$ be a given $n \times n$ matrix and let $A_0$ ...
0
votes
1answer
89 views
Orthogonal in the B Norm?
If you have two generalized
eigenvectors $\varphi_1 , \varphi_2$
(with different eigenvalues) of a
matrix A, then they will be
orthogonal in the B norm.
In this context, I do not ...
6
votes
1answer
324 views
Does convexity of a 'norm' imply the triangle inequality?
Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we ...
12
votes
2answers
2k views
Difference between metric and norm made concrete: The case of Euclid
This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.
This time I am making ...
0
votes
2answers
230 views
Matrix norm characteristics
$$\|A\|_p = \displaystyle \max_{\|x\|_p = 1} \|Ax\|_p $$
$$\|A\|_2 \leq \|A\|_F \leq \sqrt{n}\|A\|_2$$
How I can show that $1$ and $2$ are correct?
$2)$
$||Ax||_{2}=\sqrt{\sum_{i=1}^{n} ...
1
vote
2answers
101 views
Norms for space $V$ of vector-valued mappings $f: \mathbb R^m \rightarrow \mathbb R^n$
I wonder whether there are natural norms for the space $V$ of vector-valued functions that map $\mathbb R^m$ into $\mathbb R^n$.
Formally, let's define $V$ as the set of $f$ such that $f: \mathbb R^m ...
8
votes
4answers
2k views
How do I calculate the $p$-norm of a matrix?
I know that the $p$-norm for a matrix is:
$$\|A\| = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}$$
but I don't know what this really means.
So how would I compute the $2$-norm, $3$-norm, etc for the ...
1
vote
0answers
100 views
How to define an inner product for normed space? [duplicate]
Possible Duplicates:
Norms Induced by Inner Products
From norm to scalar product
Given a normed space $(X, \|\cdot\|)$. Is it possible to define an inner product $\langle \cdot, \cdot ...
4
votes
1answer
3k views
What is the difference between the Frobenius norm and the 2-norm of a matrix?
Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?
If they are ...
7
votes
2answers
245 views
Is a completion of an algebraically closed field with respect to a norm also algebraically closed?
Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
0
votes
1answer
301 views
Ky Fan Norm Question
How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix)
Thanks.
0
votes
1answer
119 views
parametrize hypersphere
I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$.
Is there a general parametrization of $p$-norm hyperspheres ...
4
votes
1answer
211 views
Prove a basic fact on a linear combination of vectors
Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact?
Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
9
votes
2answers
660 views
How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
I want to prove the following theorem (no idea whether it has a name):
Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
2
votes
1answer
398 views
Projecting onto vector space with L-Infinity norm/ minimum absolute value polynomial fitting
I'm looking for a way to project a vector (in this case a function on the real line) onto a basis for that space (in this case the set of N-degree polynomials over the domain of a closed interval) ...
36
votes
3answers
4k views
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ || \cdot || $.
It's not hard to show that if $|| \cdot || = \sqrt{\langle \cdot, \cdot \rangle}$ for some ...
7
votes
2answers
413 views
From norm to scalar product
In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question ...
9
votes
2answers
2k views
Understanding the properties and use of the Laplacian matrix (and its norm)
I am reading the wikipedia article on the Laplacian matrix:
http://en.wikipedia.org/wiki/Laplacian_matrix
I don't understand what is the particular use of it; having the diagonals as the degree and ...
6
votes
2answers
735 views
Why should I avoid the Frobenius Norm?
I vaguely remember the Frobenius matrix norm
( ${||A||}_F = \sqrt{\sum_{i,j} a_{i,j}^2}$ ) was somehow considered unsuitable for numerical analysis applications. I only remember, however, that it was ...
1
vote
2answers
260 views
Motivation for normed space definition?
So if $X$ is a vector space, and you define a norm, $x \mapsto \| x \|$, on it, then the bounded subset, $V = \{ x \in X: \|x\| < \infty \}$ is automatically a subspace. This follows from the ...
8
votes
1answer
983 views
How are norms different from absolute values?
Hopefully without getting too complicated, how is a norm different from an absolute value?
In context, I am trying to understand relative stability of an algorithim:
Using the inequality ...
4
votes
1answer
367 views
What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?
Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the ...
2
votes
2answers
411 views
Can a linear operator from a normed space to itself be extended with operator norm preserved? [closed]
Let $X$ be a subspace of $\ell_1^4$ (i.e. $\mathbb{R}^4$ equipped with the $\ell^1$ norm). Can one always extend a linear operator $l:X\rightarrow \ell_1^4$
to $L:\ell_1^4\rightarrow \ell_1^4$ such ...
2
votes
3answers
2k views
Norm of a symmetric matrix?
Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I ...
