0
votes
1answer
41 views

Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows \begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...
0
votes
0answers
47 views

Relation between the block maximum norm and the Euclidean norm of a matrix

I am trying to give answer to the following question: Let's define de block maximum norm of a $N*M \times N*M$ matrix as \begin{eqnarray*} \parallel A \parallel_b = max_{x \neq 0} [ \parallel A x ...
2
votes
1answer
81 views

Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...
6
votes
2answers
306 views

$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex vector space. As a direct consequence (or rather a particular case) of the Riesz-Thorin theorem, we have $$ \|T\|_2 \le ...
10
votes
1answer
306 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
0
votes
2answers
144 views

Matrix-Norm aquivalence with p-Norm [closed]

Let $A$ be a square Matrix and $||\cdot ||_p$ the induced Matrixnorm for $1 \leq p \leq \infty$. Is it true that $$||A||_p\leq \max(||A||_1,||A||_{\infty})?$$ For $p=2$ the answer is yes because ...
0
votes
1answer
270 views

solving trace norm equality [closed]

Problem Formulation under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$. ...
1
vote
0answers
87 views

expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$. Let $x \in ...