# 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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### Question on upper triangular matrix with complex eigenvalues with modulus less than 1

This is problem 16, Section 6.B from Linear Algebra Done Right, 3rd Edition. Suppose the field is $\mathbb{C}$, $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, all the eigenvalues of $T$ have ...
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### Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
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### Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
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### Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| \end{...
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### 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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### 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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### Proving $|x|$ is a norm in $\mathbb {R}^n$

I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$. Note that $$|x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2}$$ Thank you, Klara
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### Does $N(z)=\pm 1$ imply $z$ is a unit in $\mathbb{Z}[\sqrt{10}]$?

I've been trying to prove that $\mathbb{Z}[\sqrt{10}]$ is not factorial. I did this by defining the norm $N(a+b\sqrt{10})=a^2-10b^2$. I was able to show for myself that $N(z)=\pm 2$ and $N(z)=\pm 5$ ...