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

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Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
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253 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
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293 views

Need help understanding matrix norm notation

I've been trying to understand matrix norms (full disclosure: school assignment, not looking for answers, just clarity!), and how they follow from vector norms - been awhile since I did much linear ...
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127 views

What are the maximumnorm and supremumnorm of a vector when having a basis?

I have a perhaps stupid question. When having a finite-dimensional Vectorspace $X$ (f.e. n-dimensional) and when knowing a basis $V=\left\{v_1,...,v_n\right\}$ of it, so any $x\in X$ can be written as ...
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1answer
140 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
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258 views

Derive steepest descent vector subject to norm constraint

I am currently working through an old textbook Practical Optimization by Gill, Murray and Wright (c 1982) who make some derivations which seem correct, but I am unable to duplicate. In the equations ...
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58 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
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441 views

Proof of a norm

Prove that: $ \|\cdot\|_1:\mathbb{R}^n \rightarrow \mathbb{R};\vec{x} \mapsto \sum_{j=1}^n |x_j| $ is a norm defined on the vector space $\mathbb{R}^n$. 1) Zero vector: $\sum_{j=1}^n |x_j| = |x|_1 ...
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90 views

Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
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1answer
132 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
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Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
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1answer
211 views

p norm Matrix relationship

I am trying to show that $\Vert A \Vert_\infty \leq \sqrt{n}\Vert A \Vert_2$ given that $A \in \mathbb{R}^{m \times n}$, $\Vert A \Vert_\infty = \max \limits_{1\leq i \leq n} \sum\limits_{j=1}^n ...
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206 views

equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
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78 views

Norm of Polynom

I'm trying to understand the following equation: $$\langle p,q\rangle_p=\int_{-1}^1p(x)q(x)dx\\ \text{Basis: }\{p_1,p_2\}\\ p_1:=2x,\quad p_2:=x-1$$ Norm: $$p_1:q_1=\frac{p_1}{\|p_1\|_p}$$ ...
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375 views

What is the reason norm properties are defined the way they are?

Suppose we have a complex vector space $V$. A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies (i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity) (ii) $f(x + y) \le ...
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402 views

Poincaré's lemma with norm in $H_{0}^{1}$

I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
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228 views

Equivalence of Norms Defined on a Cartesian Product

While studying some notes on normed vector spaces, I have come upon the proof that addition $+:V \times V\to V$ of vectors in a normed vector space $V$ is a continuous operation. The proof of this ...
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1answer
231 views

Dual norm and distance

Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$. I would like to find a function $f\in X^*$ that satifies $||f||\le1$, $f(x)=d$ and ...
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1answer
167 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
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Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$

With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} $$ ...
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146 views

Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
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1answer
262 views

Is there any main method for finding norm of function in $L_1$ space?

Is there any main method for finding norm of function in $L_1$ space? For example : $f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$
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57 views

Single norm criterion

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
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362 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
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114 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
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1answer
54 views

Prove that absorbing sets induce quasinorms

I have been trying to prove this for so long that I pose this to Math.SE: Let D be a bounded (in regards to, for example, the maximum norm), absorbing subset of a finite-dimensional vector space V ...
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0answers
236 views

Why is the subdifferential of norm of a matrix ||A|| defined like this?

I read in a paper called "Characterization of the subdifferential of some matrix norms" that it defines the subdifferential of the matrix norm like this: $$\partial ||A||=\{G \in R^{m\times n} : ...
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116 views

norm of a matrix ( which norm have to use ?)

I need to find the norm of the matrix $$ A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\ -e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right) $$ Here ...
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Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
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43 views

Question about norms

Denote supnorm by $|x| = \max\{|x_1|,...,|x_n|\}$ where $x \in \mathbb{R}^n$. How can we show that this norm and Euclidean norm satisfies the following inequality? $$ |x| \leq ||x|| \leq ...
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119 views

Small question regarding norms and Holder conjugates.

I'm trying show that if $p,q$ are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$ Where ...
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345 views

Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$. Suppose we have matrices $M$ and $S$ with leading ...
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The difference between $L_1$ and $L_2$ norm?

I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out. In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down ...
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Norm of vector in $\mathbb{R}^3$ with multiple

If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then: May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = ...
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1answer
106 views

Matrix norm relationship

Suppose we have two matrices, A and B, and $\left\Vert A\right\Vert _{F}\geq\left\Vert B\right\Vert _{F}$ where $\left\Vert .\right\Vert _{F}$ denotes Frobenius norm. Does it imply $\left\Vert ...
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Linear Algebra munkres analysis on manifolds question.

If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then $$ |AB| \leq m|A||B|$$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$ Attempt: $ |AB| = \max\{| ...
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Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
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Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
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Why the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} :
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721 views

Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
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How to represent and handle eigenvalue constraint in optimization

For example consider the problem $\min_X f(X)$ s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$ where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$ ...
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is product of norms convex?

Is a function of the form $f(x) = \|x\|_1\|x\|_2$ convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
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convergence of $L^p $ norm [duplicate]

Possible Duplicate: Limit of $L^p$ norm If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
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Norm properties and completeness

Let $(X,||.||_X)$ be a normed space, M,N two subspaces with norms $||.||_M,||.||_N$ The identity maps are cont. Now I can define the norm $||x||_{M+N}=inf\{||m||_M+||n||_N:m\in M, n\in N, x=m+n\}$ ...
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231 views

Quotient norm on $X\backslash M$

I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions: 1) Is the quotient norm a norm on the quotient space X\M ? What I ...
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251 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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151 views

Distance between point and linear Space

Suppose $E$ is a normed vector space. Let $f$ be a continuous linear functional on $E$ and denote by $M$ the Kernel of $f$. Let $x\in E$. How to show that ...
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183 views

convergence in $L^2$

Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$. Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty ...
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why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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703 views

Equivalence of norms is a equivalence relation

Two norms $||-||_1 $, $||-||_2$are equivalent if: for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$ This is a ...