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Questions tagged [norm]

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

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Show that $\|b \|^2\ (\sin\theta)^2 = \min_x\ \|ax-b\|^2$ where $\theta$ is the angle between $a,\ b$.

Show that $\|b \|^2\ (\sin\theta)^2 = \min_x\ \|ax-b\|^2$ where $\theta$ is the angle between $a,\ b$. This is for vectors $a, b \in \mathbb{R}^m$ \ {$0$}. The second part of the question says: ...
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How to prove $|x_i - x_j| \geq |y_i - y_j| - 2 ||x - y||_\infty $

Let $x$ and $y$ be two real vectors of length $n$. Let subscripts $i$ and $j$ denote the $i$-th and $j$-th elements of a vector. How can we prove $$|x_i - x_j| \geq |y_i - y_j| - 2 ||x - y||_\infty $...
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Uniform limit of complex difference quotient

Let $\mathcal{C}_0([0,1])$ be the set of continuous complex-valued functions defined on $[0,1]$ such that they vanish at the origin and let $\|\cdot\|_{MAX}$ denote the norm $$\|x\|_{MAX} = \max_{t\...
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Operator norm bounds

For $A \in \mathbb{R}^{m \times n}$m need to prove the following bounds: \begin{align*} ||A|| &\leq \sqrt{m} \max_{i \in \{1, 2, \dots, m\}} \big( \sum_{j=1}^n A_{ij}^2 \big)^{1/2}\\ ||A|| &\...
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Clueless on how to prove $\|(a,b)\| \le |a|+|b|$ [duplicate]

Basically, the inequality says that the norm of a vector is always less than or equal to the sum of the absolute value of its components. And I know that the norm is defined as: $$\|(a,b)\|=\sqrt{a^...
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orthogonalization and orthonormalization

I have to construct a diagonalizable and orthogonal matrix starting from this quadratic form $Q(x_1,x_2,x_3)=-2x_1x_3+2x_1x_3-2x_2x_3$ in order to reduce it in canonic form with a variables change. I'...
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dual relationship $\|x\|_\infty = \max_{\|z\|_1 \le 1} z^T x$

I was reading the answer in the stackexchange question about dual cones of L-1 norm cone It says the key is the dual relationship $\|x\|_\infty = \max_{\|z\|_1 \le 1} z^T x$. I am trying to ...
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Intrinsic curvature and is the space described by polar coordinates Euclidean?

I am trying to get my head around more formal concepts in geometry for the purpose of understanding gradient descent and natural gradients in machine learning. But I feel I still do not understand ...
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What does it mean when the map $u \mapsto \lVert u \rVert$ is continuous with respect to the norm $\lVert u \rVert_1$

Given any norm $\lVert \, \rVert$ on a vector space of dimension $n$, for any basis $(e_1, \dots, e_n)$ of E, observe that for any vector $x=x_1 e_1 + \dots x_n e_n$, we have $$\lVert x \rVert = \...
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Scaling cumulative path length. Scaled sum of norms.

I have a list of $n$ 2D vectors that are segments of a line path. $$v_0, v_1, v_2,..., v_n$$ Define the cumulative path length $l_x$ up to the vertex $v_x, 0 \le x \le n$ in the path as the sum of ...
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Possible norms of transformed vector with initial length of 1

Assuming we have a vector with dimension 5x1 of unit length and we change its dimensions by using with m x 5 sized matrices. Example: $$ v= \left[ \begin{array}{ccc} v_1\\ v_2\\ v_3\\ v_4\\ v_5\\...
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Show that $\sum_{l=0}^{\infty}(\gamma\matrix{A})^l$ converges

Based on the following conditions where $\matrix{A}$ is a matrix: $\left| \lambda\matrix{A} \right|=\left| \lambda \right|\left| \matrix{A} \right| $ for any $\lambda \in \mathbb{R}$ $\left| \matrix{...
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Defining a norm on Vector space

Let V be a $\mathbb{R}$-Vector space with inner product $\langle •,• \rangle : V\times V \to \mathbb{R}$ Notice $||v||=\sqrt{\langle v,v \rangle}$ Define a norm on V$$ What are they ...
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Why Do We Only Take Norms Over Real/Complex Numbers?

By definition, norms are defined over some $\mathbb{R}$ or $\mathbb{C}$ vector space. Why do we only restrict ourselves to these fields when other fields give rise to interesting objects as well? (e.g....
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Norm of a matrix and its conjugate.

If $A$ is a $m \times n$ matrix and $p, q \geq 1$ are conjugates satisfying $\frac{1}{p}+\frac{1}{q} = 1$, then prove that $$\|A\|_p=\|A^t\|_q$$ where $A^t$ is the transpose of $A$. Any hint or ...
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Prove that if $||Tu-Tv||=||u-v||$ then $T$ is of the form $Tu=p+Au$ with $A$ linear.

Consider the norm $||u||=|x|+|y|$ if $u=(x,y)$. Prove that if $T:\mathbb{R^2}\longrightarrow{\mathbb{R^2}}$ satisfies $||Tu-Tv||=||u-v||$ then $T$ is of the form $Tu=p+Au$ with $A$ linear. I've been ...
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Algebraic derivation of a formula with norms

Assume that $x_{n}$, $\widetilde{x}_{n}$ and $\bar{x}$ are column vectors and $u_{i}$ are orthonormal basis vectors. When considering equations (1) and (2), it is not clear to me how equation (3) ...
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Operator into the dual space is compact

I want to solve the following Let $X,Y$ be Banach spaces, with compact embedding $X\hookrightarrow Y$. Define the bilinear form $b:X\times Y\to\mathbb{R}$ that satisfy $$b(u,v)\leq C\|u\|_X \|v\|...
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Chain rule for derivative of a norm

Suppose that $A$ is an $M \times N$ matrix, $x$ is an $N \times 1$ vector and $b$ is an $M \times1$ vector. I want to compute $\frac{d}{dx}||Ax+b||^2_{2}$. According to this link, the answer should ...
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the acceleration of nuclear norm of similar sub-problems, perhaps SVT?

i have come up with a model with two sub-problems. It looks like: $$ ||A||_* + \lambda_1||A-G-H_1||_F^2 $$ $$ ||B||_* + \lambda_2||B+G-H_2||_F^2 $$ So it involves two independent problems of solving ...
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Derivative of Euclidean norm

Assume $X$ is a $n$ by $d$ matrix, $\alpha$ is a $n$ by $1$ vector, then $$\frac{d\|X^T\alpha\|^2_2}{d\alpha}=\frac{d\|X^T\alpha\|^2_2}{dX^T\alpha}\frac{dX^T\alpha}{d\alpha}=2\alpha^T X X^T.$$ I was ...
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Find Infinity Norm of a polynomial

I have a polynomial of known degree $n$ with initial numerical coefficients $a$ set to $0$ or $1$ such as: $p= a_nx^n +a_{n-1}x^{n-1}+...+ ax +a_0$ I have to perform a series of computations by ...
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Bounded matrix norm

This might be obvious, but I cannot prove it right away nor find a reference. Consider a matrix $A\in\mathbb{R}^{m\times n}$, $m>n$ having full column rank and a set $X$ of vectors $x\in\mathbb{R}^...
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Bounding a sup in a continuous space using the l_infinity norm

Suppose I have two functions $f(x): x\mapsto \mathbb{R}$ and $f^\prime(x): x\mapsto \mathbb{R}$. Suppose that $x \in [a,b]$ is a continuous space between two positive numbers $a$ and $b$. Say I ...
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$\sup \left\| A x + B y + C z \right\|$ subject to $\left\|x\right\| = \left\|y\right\| = \left\|z\right\| = 1$

I'm interested in finding $\sup \left\| A x + B y + C z \right\|$ subject to $\left\|x\right\| = \left\|y\right\| = \left\|z\right\| = 1$ where $A$, $B$, $C$ and $x$, $y$, $z$ are real matrices ...
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Deriving an upper bound for a constrained infinity norm minimization problem

I have the following problem: $$\min_{x\in X}\|Mx-c\|_{\infty}$$ I am considering a particular case, in which: $$M=\left[\begin{array}{cc} a & 1-a\\ b & 1-b \end{array}\right], c=\left[\...
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Using the Binomial Theorem to expand the magnitude of the difference of two vectors

I have the following expression that I need to expand using the Binomial Theorem: $$\frac{1}{\mid\vec{r}-\vec{d}\mid}$$ Now the Binomial Theorem is the following: $$(x+y)^r = \sum^{\infty}_{k=0}\binom{...
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Bounding the infinity norm using the l_2 norm

Suppose $\Big(\sum_{k =1}^n x_k^2\Big)^{1/2} \leq \eta_2$. $\eta_2$ is the bound on the l_2 norm of a matrix. I want to upper bound $\eta_\infty = \max_{i}|x_i|$ using $\eta_2$ the bound on the l_2 ...
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Norm of identity operator

Let $1<p,q<\infty$, so I need to find a norm of identity operator $J: l^n_p \to l^n_q$ When $p\le q$, it's quite easy for me to understand that $||J||=1$ (beacuse in this case $||x||_q \le ||x||...
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L1, L2 , and L infinite-norms for the size of a signal [closed]

Calculate the L1, L2 , and L infinite-norms of the following finite-length sequences {x1[n]}= {0.92 2.34 3.37 1.90 -3.59 -0.75 3.48 3.33}, where x[0] = 1.90. I use the function for Finite ...
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How to find Analytic solution of minimization of Frobenius norm?

min $ {{\|X_{t}-D^{s}{A_{t}}^{s}-{D^{r}}_{t}{A_{t}}^{r}\|}_{F}}^{2}$ Derivative of above equation with respect to ${A_{t}}^{r}$equals to $0$ gives us the analytic solution as, ${A_{t}}^{r} = {({{D^{r}...
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Using SVD to express the normalized norm of the output of a linear map

This is probably trivial, but it's been quite some time since my Linear Algebra exam, and my SVD skills are rusty. 1) Let $T$ be a linear map from $\mathbb{R}^n$ to $\mathbb{R}^n$, and $M$ the ...
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Generalized Pythagorean theorem application

let $S$ be bounded piece of a plane in the space $E_3$ and let's note $S_i$ an orthogonal projection of $S$ into $xy$, $xz$ and $yz$ planes respectively. Then it can be proved that: (1) $\text{area}(...
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Properties of the uniform norm on probability spaces

Let $(X, \mathcal A, \mu)$ be a probability space, and $v\in \mathcal M^+(\mathcal A)$ be a positive measurable function. I am interested in how to show that $$ \left| \int_X v ~ \text d\mu \right| \...
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Inequality between norms in $L^p$ and $L^q$ spaces for $p,q>1$

If I define $2$ continuous probability density functions $f(x,y)$ and $g(x,y)$ for which $L^p$ and $L^q$ norms are defined for $p>1$ and $q>1$, Is it correct to say that $\left\Vert f\right\...
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Inequality involving supremum norm on integral

Given is that $T:C([0,a])\rightarrow C([0,a]),\space (Ty)(x)=\frac{x^2}{2}+\int_{0}^{x}ty(t)dt, \space||y||_1= sup_{x \in[0,a]} |y(x)|$. I want to prove that $||Ty-Tz||_1\leq \frac{a^2}{2}||y-z||_1$. ...
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Is it true that $\|A\|_\infty \le \sqrt{n}\|A\|_2$

Let $A\in \mathbb{C}^{m,n}$ be a matrix. Show that that $\|A\|_\infty \le \sqrt{n}\|A\|_2$. I'm wondering, however, if this statement is actually true at all. Could it be that the $\|\cdot\|_\infty$ ...
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Prove ||u-v|| defines a metric space

Given a normed linear space, I need to prove that $||U-V||$ defines a metric space. I can prove symmetry and positivity, but having trouble proving $||U-V||+||V-W||\geq ||U-W||$ I know : $$||U-W||\...
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Integral inequality in a probabilty space

Let $(X,\mathcal{A},\mu)$ be a probability space. Let $v \in \mathcal{M}^+(\mathcal{A})$. Show that $|\int_X v \, d\mu | \leq ||v||$. $||v||$ denotes the uniform norm. I know the following but ...
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Fixed point of non-linear system involving the L2 norm?

My apologies for the not-very-descriptive title - any suggestion on how to phrase this question is welcome as this would make googling for approaches much easier. Is there a closed-form solution to ...
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It is banach space or just normed space ? [duplicate]

Let $X$ be a nonempty set and $B(X,C)$ the set of all complex functions defined on $X$ and $\sup|f(x)|<+\infty$. Define norm $||\cdot||$ on $B(X,C)$ by $||f||=\sup|f(x)|$. Is $(B(X,C),||\cdot||)$ ...
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Normed spaces where $x, y \neq 0$, $\Vert x + y \Vert = \Vert x \Vert + \Vert y \Vert $ and $\forall c > 0, \ x \neq cy$

For vectors $x, y \in \mathbb{R}^n\setminus\{0\}$, under the Euclidean norm we have that $$\left\Vert x + y \right \Vert_2 = \left \Vert x \right \Vert_2 + \left \Vert y \right \Vert_2 \iff \exists c \...
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Do the two linear operators on $M_n(\mathbb R)$ have the same induced norm on the subspace of symmetric matrices?

Let $A \in M_n(\mathbb R)$ be fixed with spectral radius $\rho(A) < 1$. Then $T_1, T_2$ are two well-defined linear operators on $M_n(\mathbb R)$ given by \begin{align*} T_1(X) = \sum_{k=0}^{\infty}...
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How can I derive the expression of sensitivity of logarithmic norm (matrix measure)?

The different logarithmic norm of a matrix is defined as: $$ \mu_\infty(A)=\max_i \{a_{ii}+\sum_j |a_{ij}| \}$$ $$ \mu_2(A)=\lambda_\max \bigg (\frac{A^*+A}{2}\bigg )$$ Let, $A=A_o+\sum_iA_ix_i$ be a ...
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Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$?

Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$? Here $\circ$ is the Hardamard product and $\|•\|_2$ is the Frobenius norm.
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Derivative of an Euclidean norm of vectors

In his Convex Optimization book, Boyd has a proof for the Separating hyperplane theorem. In page 48 in particular there is this part that I am not sure how it is done, How did we end up from the ...
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Why is the product of two norms is always bigger or equal to the norm of the same corresponding element? [closed]

E.g. if $A$ is a matrix and $v$ is a vector which can be multiplied with the matrix, it always applies that (no matter how the norm is defined): $||A|| ||v||\geq ||Av||$ Why is it so?
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Does a norm induce an inner product uniquely in the sense that $\|x\|^2 = \langle x,x\rangle?$

Let $V$ be a normed space with a norm $\|\cdot\|.$ The polarisation identity Wiki page states that In a normed space $(V,\|\cdot\|),$ if the parallelogram law holds, then there is an inner ...
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Upper bound for $(AB-BA)x$

Given matrices $A,B\in\mathbb{R}^{n\times n}$where matrix $A$ is a diagonal matrix and $B$ is an upper triangular matrix. I'm looking for an upper bound for the expression \begin{align*} (AB-BA), \end{...
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Is the 2-norm of the consensus part of a primitive row-stochastic matrix less than 1?

Let $A$ be a primitive row-stochastic matrix. By Perron-Frobenius theorem, $A$ has an eigenvalue 1 and corresponding left eigenvector and right eigenvector $\pi$ and $\mathbb{1}$, i.e., $A\mathbb{1}=1,...