# Questions tagged [norm]

This tag is for the questions relating to the norm which is a function on a vector space $X$ that generalizes notion of length of vector in general vector spaces. A vector space $~X~$ with a distinguished norm is called a normed space.

5,505 questions
Filter by
Sorted by
Tagged with
43 views

### What is the benefit of defining a positive norm for vectors?

I read that the reason we have the property $\langle A|B\rangle=\langle B|A\rangle^*$ is to make define a positive norm with the formula $\langle A|A\rangle$. Though I do not understand how having ...
66 views

### For a normed vector space, is $\|x-y\| \leq \|x\|+\|y\|$ true?

I have a question about an inequality in normed vector spaces and I want to know if my proof is correct. Claim: Let $X$ be a normed vector space. Then \|x-y\| \leq \|x\|+\|y\|\end{...
36 views

20 views

### Scalar derivative of vector norm

Can someone check my math here? I feel like this should be a very simple problem, but I can't seem to confirm this by searching. What is the derivative of a vector norm? I'm referring to the usual ...
46 views

### Proving the triangle inequality for a function on matrices

Let $V$ be a $m$-dimensional vector space over the set of $n \times n$ matrices. For instance a vector would be $$v=(M_1,M_2,\dots, M_m)$$ where each $M_i$ is a $n\times n$ matrix. I now define ...
21 views

### Does $T_n$ converge to the identity operator $I$ by the operator norm in $C[0,1]?$

Consider a sequence of operators $T_n$ :$C[0,1]\rightarrow C[0,1]$ given by the formula $$(T_nx)(t)=x(t^{1+\frac{1}{n}}),\ t\in [0,1], \ n\in \mathbb{N}.$$ Does $T_n$ converge to the identity operator ...
26 views

### Inner product of vector of matrices - I have the norm

Let G be the set of all 4 x 4 matrices with positive determinant. I define v as a vector such matrices. The norm of v is defined as: $$||v||^2=\sqrt{\sum_{g \in v}\det g}$$ What is the inner product?...
46 views

### Are all matrix norms equivalent?

I have shown that in $\mathbb{R}^n$all vector norms are equivalent. Does the same hold also for all matrix norms $\|\cdot \|:\mathbb{R} ^{n\times n} \rightarrow \mathbb{R} ^2$?
20 views

### Prove that the Frobenius norm is invariant under orthonormal projection

Assume I can express a rank-deficient, $N\times N$ symmetric covariance matrix $\Sigma$ as $$\Sigma=\mathbf{USU}^\top$$ where $\mathbf{U}$ is an $L\times N$ orthonormal ...
36 views

### Norm of a vector component (considering an orthogonal basis) is always lesser than or equal to the norm of the entire vector in $\mathbb R^n$ [closed]

Suppose $\mathbb R^n$ (with the usual dot product as inner product) is equipped with some arbitrary norm $||\cdot||$. Now let's say $\mathbf x = (x_1, x_2, \ldots, x_n)$ is a vector in $\mathbb R^n$ (...
19 views

### Matrix Derivative of F-norm with Hadamard Product

I'm trying to solve $\nabla_X \| A \odot(B-X^\top C) \|_F^2$, but I don't know how to solve this... Could anyone help? Thank you in advance for any help you can provide.
22 views

### Bounded linear functional on a Hilbert space.

Let $H$ be a complex Hilbert space with orthonormal basis $\{e_n,n=1,2,\cdots\}$ and $f$ be a linear functional on $H$ defined by $$f(x) =\sum_1^\infty \langle x, e_n\rangle \frac{1}{n}.$$ How to ...
60 views
+50

### Minimize $x^*(A+A^*)x$ such that $x^*A^*Ax=1$ and $x^*x=1$

Given $A\in\mathbb{C}^{n\times n}$, such that it has singular values larger than $1$ and smaller than $1$, \begin{array}{ll} \underset{x\in\mathbb{C^n}}{\text{minimize}} & x^*(A+A^*)x.\\ \text{...
12 views

42 views

32 views

18 views

### Real, symmetric matrices $A, B \ge 0$ with $A$ positive definite, must we have $\lVert (A +B)^{-1}B \rVert_{\text{op}} \le 1$?

If $A$ is positive definite and $B$ is positive semi definite, both symmetric real matrices, with $A - B$ positive semi-definite, must we have $\lVert (A+B)^{-1}B \rVert \le 1$ in the operator norm? I ...
68 views

### Norm inequality for positive semidefinite matrices

Suppose $A, B\ge0$ are positive semidefinite matrices on the complex field, is it true that $$\Vert A^2 +B^2 \Vert \le \Vert A + B \Vert^2,$$ for the spectral norm? I have tried numeric tests and the ...
Can someone help me to provide an upper or lower bound to the following? $$\|\nabla_xf(x_1)\|^{\alpha}-\|\nabla_xf(x_2)\|^\alpha\leq?$$ where $0<\alpha<1$. I prefer a bound in terms of $\|\... 1answer 42 views ### Show that the operator norm$\sum$defined as$\sum(\lbrace{x_n\rbrace}_{n=0}^\infty) = \sum_{n=0}^\infty \lbrace{x_n\rbrace}$is unbounded. Suppose$\|\lbrace{x_n\rbrace}_{n=0}^\infty \|_{\text{sup}} = \text{sup}\lbrace{|x_n|\rbrace}_{n=0}^\infty $. Let$\sum$be a linear operator from the$A \to \mathbb{R}$defined as$\sum(\lbrace{x_n\...
Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.) For $A\in \mathcal{C}$ ...