1
vote
1answer
28 views

Why are the spectral norm of $A^{*}A$, $AA^{*}$ and $A$ equal?

I'm learning matrix norm now, but i don't have learned Hermitian before. Is there any theorem about hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
1
vote
1answer
31 views

Can you divide by matrix norms in an equation?

Supposing that a matrix A has an eigenvalue lambda, show that for any induced matrix norm, $||A|| \geq |\lambda|$. I attempted the solution, but I am not sure if it is valid to cancel the norm of ...
2
votes
1answer
32 views

$\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$

I want to prove this $\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$\ Suppose $\lambda_0>\lambda_1>\dots>\lambda_{n-1}\ge 0$ are distinct eigen values of $T^*T$ and ...
2
votes
0answers
50 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
0
votes
1answer
34 views

when norm of an operator is given by max of eigen values modulas

Could any one tell me how this $\|x\|^2=\|x*x\|$ and the rest of it? I know $\|x\|=\|x^*\|$, I also understand $x^*x$ is hermitian and so diagonalizale but then did not understand the norm square ...
0
votes
4answers
101 views

how to interpret this norm

Could any one tell me what "x" is there when he has defined $\|x\|$, just after he says $M_n$ has the operator norm thank you for helping.
2
votes
1answer
46 views

Proof that the condition number of an isometry matrix is 1

At first glance it seems somewhat trivial, but I have some doubts, so I'd like your opinion. We are given that $\left\|Ax\right\| = \left\|x\right\|, ~ \forall x \in \mathbb{C}^{n}$ and want to show ...
1
vote
1answer
40 views

Inequality for subordinate norm of a submatrix : $\Vert B\Vert\le\Vert A\Vert$.

We 'did' this exercise in class Show that the subordinate norm of a submatrix is smaller than the the subordinate norm of the matrix i.e. $$\Vert B\Vert\le\Vert A\Vert$$ where $B$ is the sub ...
-1
votes
1answer
10 views

Prove the following version of the polarization identity:

I need to show $<x,y> = \frac{1}{2 \pi} \int_{-\pi}^\pi ||x + e^{i \theta}y||^2 e^{i \theta} d\theta$ where $<x,y>$ is inner product. So far I have: $\frac{1}{2 \pi} \int_{-\pi}^\pi ||x + ...
0
votes
1answer
40 views

Subordinate norm is equal to $\Vert A\Vert_M=\Vert MAM^{-1}\Vert$ for the norm $\Vert x\Vert_M=\Vert Mx\Vert$

Let $M\in \mathcal{M}_n(\Bbb{K})$ a nonsingular matrix, we have $\Vert x\Vert_M=\Vert Mx\Vert$ is a norm over $\Bbb{K}^n$. I have to prove that the subordinate norm is equal to $\Vert A\Vert_M=\Vert ...
0
votes
3answers
133 views

Is it true that $\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1$?

I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension ...
0
votes
1answer
65 views

Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
0
votes
1answer
44 views

What is a good reference for learning about induced norms?

Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces". Which makes me want to ask: what's a good ...
0
votes
1answer
53 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
1
vote
0answers
60 views

Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
2
votes
1answer
49 views

Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by ...
1
vote
1answer
75 views

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
0
votes
1answer
70 views

Proving result on matrix norms

How do I prove that, letting, for $A\in\mathbb{C}^{n\times n}$: $$(a)\quad\|A\|_1=\max\limits_{i=1,\cdots,n}\left(\sum\limits_{j=1}^n|a_{ij}|\right),$$ $$(b)\quad\|A\|_2=\rho^{\frac12}(A^HA),$$ with ...
2
votes
1answer
54 views

Matrix inner product, and operator and trace norm inequality

I have trouble proving the following inequality. Let a matrix $A \in \mathbb{R}^{M \times N}$, and $\sigma_i(A)$ be the i-largest singular value of A. Define the operator norm and the trace norm as ...
3
votes
1answer
57 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
0
votes
1answer
58 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
1
vote
2answers
63 views

Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
2
votes
1answer
83 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
1
vote
1answer
28 views

Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
2
votes
1answer
21 views

Is $||A||_F ||x||_2^2 \geq x^TAx$

Given a symmetric matrix $A$ and a vector $x$ Is $||A||_F ||x||_2^2 \geq x^TAx$? If yes, how to show this?
2
votes
1answer
25 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an ...
1
vote
1answer
35 views

Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
2
votes
1answer
40 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
3
votes
2answers
80 views

Norm of Hilbert matrix is it equal to $\pi$?

Let $A$ be a Hilbert matrix, $$a_{ij}=\frac{1}{1+i+j}$$ We have the following result : $\Vert A\Vert\leq \pi$. I am using the subordinate norm of the euclidean norm i.e. $$ \Vert A\Vert=\sup\{\langle ...
2
votes
1answer
20 views

Operator in $\mathbb R^2$

I am a bit confused, can someone help me with the following? Is there an operator $T$ in $\mathbb{R}^2$ such that: $\parallel u \parallel +\parallel v\parallel = \parallel T(u+v)\parallel$ for every ...
1
vote
1answer
48 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
3
votes
1answer
116 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
2
votes
2answers
55 views

Condition for a norm be absolute

Let $\|x\|_B\mathrel{\mathop:}=\sqrt{x^{t}Bx}$, where $B \in \mathbb{R}^{n\times n}$ is a symmetric and positive semidefinite matrix. If $\mid x\mid = (|x_1|,|x_2|,\ldots,|x_n|)$, I want to show that ...
2
votes
1answer
59 views

Proof that frobenius norm is a norm [duplicate]

It's pretty basic and I'm sure I'm missing something dumb here, but I'd like to know why $||A+B||_F \leq ||A||_F+||B||_F$ The way I understand it, ...
2
votes
1answer
95 views

Is the matrix least squares minimizer (Frobenius norm) the same as the matrix 2-norm minimizer?

Given matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times k}$, consider the (least squares) minimizer $\arg \min_{X \in \mathbb{R}^{m \times k}} \| AX - B \|_F$, where $\| M \|_F$ ...
2
votes
1answer
24 views

Spectral norm of a matrix obtained by setting some entries to zero

For example can we say, that if $A$ is original matrix and $A'$ obtained from $A$ by zeroing some elements then $\|A\|_2 \geq \|A'\|_2$?
0
votes
1answer
12 views

Is there relations between earth mover's distance and vector norms?

Say I have two vectors $a$ and $b$. Can I estimate $\mbox{EMD}(a,b)$ via some combination of things like $\|a-b\|_p$ and such?
1
vote
2answers
62 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
1
vote
1answer
71 views

Upper bound on the inverse of a Grammian matrix

I have been trying to find a reasonable upper bound on the following: Given $n\in N$ and the Grammian matrix $A_n$ = (($f(i)$ , $f(j)$)) , $f(\lambda) = e^{\lambda t}$ for $0\le t \le 1$ and ...
1
vote
4answers
57 views

I need help with a proof showing $\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 $

So, I am dealing with the 2-norm and the projection is defined as the standard orthogonal projection, so far I have $$\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 ...
0
votes
1answer
37 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
votes
1answer
58 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
2
votes
1answer
40 views

Completely multiplicative matrix norm for certain semigroups of matrices.

I am currently working on some properties of matrix products and their norms for $\mathbb{R}^{n \times n}$ matrices and i was wondering if there exists a completely multiplicative matrix norm, i.e. ...
1
vote
1answer
27 views

Prove $||A||_2 = max_{x \neq 0, y \neq 0}\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}\}$ for $A\in \mathbb{K}^{n\times m}$

I'd like to prove that the spectral norm of a matrix that is not necessarily square can be written as the following subordinate norm $||A||_2 = max\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}, y ...
12
votes
1answer
329 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
1
vote
1answer
50 views

Inequality of Weighted norm

I have a question about the weighted norm inequality: The weighted norm of a vector $x\in R^{M\times N}$ is defined by: $\left \| X\right \|_{w,*} = \sum_{_{i}}\left |w_{i}\sigma _{i}\left ( X ...
1
vote
0answers
104 views

Showing $\| Mx \|^2 = x^TM^TMx$

So, I am trying to prove $$\|Mx\|^2 =x^TM^TMx,$$ however I am running into some difficulties. Here, $M \in \mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$. I know that when you take the transpose ...
0
votes
1answer
49 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
0
votes
1answer
91 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
0
votes
1answer
28 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...