Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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27 views

upper bound on a matrix norm

what is the smallest upper bound for the following norm $\|\left(\lambda\ I +A\ A^T\right)^{-1}\|<?$. where, A is a rectangular matrix, $\lambda>0$ is a scalar. (any possible norm)
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0answers
13 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
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0answers
6 views

Hilbert transform of the product of functions

Let $H[g]$ denotes a Hilbert transform of function $g$. What would be the constant $C$ in the following inequality: $$ \|H[(\cos n)(\cos{1/(2n))}f](x)\|_{L_2}\leq C\|f\|_2? $$
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14 views

Norm of arbitrary constant

I'm sitting in front of an exercise (basics in quantum mechanics), which wants me to check if the integral of a given function can be normed. One of those functions is the integral of zero from ...
3
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1answer
32 views

Is it true that $\|\text{diag($\pi$)} P\|_2 \leq 1$ for $P$ stochastic and $\pi P = \pi$.

$\| \cdot \|_2 $ is the matrix norm induced by $L_2$. $P$ is any given real square $n \times n$ non-negative matrix with rows summing to one, i.e. $P1 = 1$, where $1$ is the vector of ones. There is ...
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1answer
11 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
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2answers
59 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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1answer
25 views

Prove that this is a norm?

I have a question: I know the requirements of being a norm(the 3 requirements).I try to use them but,I don't know how to do.Can I get a litle help? Thank you.
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0answers
16 views

Prove $ \left \| \left [ G \: \: \: \: I \right ] \right \|^{2}_{\infty }\leqslant \left \| G\right \|^{2}_{\infty } +1 $

Consider the Strickly proper transfer function of G(s) and impulse response G(t), by assuming the input signal u(t) having the bounded $\left \| \right \|_{1}$ , How can one prove the following ...
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1answer
24 views

Is this a valid operator norm?

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm? (I think it is. As it satisfies ...
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1answer
55 views

Calculating a Matrix Norm

I'm trying to calculate some norm for a matrix $A = [3, 2; 0,1]$ given the formula $\|A\| = \max_{|v|=1}|Av|$, where $|v|$ is taken to be the Euclidean norm for a vector, i.e. the standard distance ...
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1answer
39 views

distance between solutions in a convex optimization

Assume that you have the following convex optimization problem: $\min_{M} \|b+A\ M\ v\|_2$ subject to : $\|M\|_{2}<1$ (maximum singular value less than 1) where M is a suare matrix (n by n), A ...
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1answer
39 views

matrix convex optimization

How to solve the following problem explicitly? I mean closed form solution if possible. $\min_{M} \|M\ a-b\|_2$ subject to : $\|M\|_{\infty}<1$ (maximum singular value) where $M$ is a square ...
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1answer
23 views

Is $\phi$ a norm of E?

Let $(E, \| \|)$ be a normed space. We define $\phi:E \rightarrow [0,\infty)$ as follows: $$\phi(e)= \dfrac{\|e\|}{1+\|e\|}$$ Is $\phi$ a norm of $E$? Please help! Thank you! P.S. This question ...
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1answer
39 views

Is this sufficient for continuity?

Assume you have a map $\phi : V \rightarrow \mathbb{C}$, where $V$ is a complex vector space. Now, if we have $\phi(\lambda x) = |\lambda | \phi(x)$ and $\phi(x+y)^2+ \phi(x-y)^2 = 2\phi(x)^2 +2 ...
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0answers
80 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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1answer
27 views

Estimate the Cauchy integral for matrix-valued functions

Recently, I became familiar with the concept of the "matrix function via Cauchy integral", i.e., $$f(A):=\frac{1}{2\pi i}\int_\varGamma f(z)(zI-A)^{-1} \mathrm{d}z$$ Furthermore, it can be shown that ...
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1answer
24 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
4
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1answer
56 views

(still open) For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
4
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0answers
51 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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0answers
13 views

Norm of product of two matrices

Let $A\in\mathrm{R}^{n\times n}$ and $B\in\mathrm{R}^{n\times n}$ be two matrices. If $\|\cdot\|$ denotes the matrix norm, are the followings true? $\|AB\| = \|BA\|$ $\|A^2\| = \|A\|^2$ If they ...
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0answers
23 views

Show that $\{c \ge 0: ||Ax ||\le c ||x ||, \forall x \in X \} $ is closed , bounded from below and nonempty?

Let $A $ be a linear map from a normed linear space $X $ to a normed linear space $Y $ (both over the reals). How can I show that the set $\{c \ge 0: ||Ax ||\le c ||x ||, \forall x \in X \} $ is ...
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1answer
29 views

Well-defined $\xi$-weighted (Euclidean) norm

Suppose $\xi$ is a vector, that is used for $\parallel z\parallel_\xi$ calculation. Should every element of $\xi$ be positive, $\xi(i)>0$?
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24 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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1answer
46 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...
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0answers
19 views

Is there a single matrix norm such that for all stochastic $P$, $\| P \| = 1$?

By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$. By matrix norm I mean a norm in the vector space of ...
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1answer
36 views

Can a continuous linear form have a norm of infinity?

We know that a linear form $A \in V^{*}$ is continuous iff $$ \exists C: C\in R, ||Av|| \leq c ||v|| \forall v \in V $$ but we know too that $$ ||A|| = min\{c>0:||Av|| \leq C||v|| \forall v\in ...
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0answers
16 views

Color distribution distance

I am looking for a distance / squared distance between two color pixel distributions that would relate to the simple L2 distance when the color distribution is over one sample. The general problem I ...
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1answer
36 views

Any isometry is an isomorphism, though the converse is not true. [closed]

If we define a mapping $f:E \rightarrow F$, where $E$ and $F$ are normed vector spaces, then $f$ is an isometry if $f$ is a linear norm-preserving bijection, that is: $\|f(x)\|=\|x\|, \quad \forall x ...
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0answers
18 views

Proving that $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on a vector space where X is a positive-definite bilinear form.

Want to show: If $X$ is a positive-definite bilinear form on a vector space $G$ with real-valued scalars and $v\in G$, then $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on $G$. Thus far I have ...
3
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1answer
36 views

Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
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0answers
61 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
4
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1answer
41 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
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2answers
19 views

Norms — Distance Between Vectors

Which two of the vectors $u=(-2,2,1)^T$, $v=(1,4,1)^T$, and $w=(0,0,-1)^T$ are closets to each other in distance for (a) the Euclidean norm? (b) the infinity norm? (c) the 1 norm? I believe I know ...
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0answers
32 views

Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
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0answers
12 views

Equivalence of condition number from equivalence of vector norms

I must show that the equivalence of vector norms implies the equivalence of the condition number of its induced matrix norm. That is, given that for two arbitrary vector norms (+ and *) and an ...
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1answer
28 views

Why are the spectral norm of $A^{*}A$, $AA^{*}$ and $A$ equal?

I'm learning matrix norm now, but i don't have learned Hermitian before. Is there any theorem about hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
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0answers
27 views

Finding norm of functional coefficents in $C[a,b]$

Let $\{x_n\}$ denote Schauder’s basis for $C[a, b]$ and let $\{h_n\}$ be the associated sequence of coefficient functionals. Compute $||h_n||$. In young's book, a schauder basis for $C[a,b]$ ...
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1answer
32 views

Can you divide by matrix norms in an equation?

Supposing that a matrix A has an eigenvalue lambda, show that for any induced matrix norm, $||A|| \geq |\lambda|$. I attempted the solution, but I am not sure if it is valid to cancel the norm of ...
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2answers
31 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
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1answer
32 views

$\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$

I want to prove this $\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$\ Suppose $\lambda_0>\lambda_1>\dots>\lambda_{n-1}\ge 0$ are distinct eigen values of $T^*T$ and ...
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2answers
9 views

Finding the norm of a complex trigonometric function?

Given that the complex norm $|z| = 1$, how would I go about proving that $|cos(z)| \leq e$? Just a hint would be helpful.
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0answers
51 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
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vote
2answers
31 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
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1answer
21 views

Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
4
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3answers
33 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
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1answer
34 views

when norm of an operator is given by max of eigen values modulas

Could any one tell me how this $\|x\|^2=\|x*x\|$ and the rest of it? I know $\|x\|=\|x^*\|$, I also understand $x^*x$ is hermitian and so diagonalizale but then did not understand the norm square ...
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1answer
29 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
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4answers
103 views

how to interpret this norm

Could any one tell me what "x" is there when he has defined $\|x\|$, just after he says $M_n$ has the operator norm thank you for helping.
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1answer
27 views

Strengthening bound in the Euclidean space

Suppose that $z$ is a $K\times 1$ vector. Denote the components of $z$ as $z_1,\ldots, z_K$. Let $r>0$ be given. I'd like to find the smallest constant $C$ such that $$ |z|^r\leq ...