0
votes
2answers
42 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
7
votes
4answers
114 views

How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
6
votes
3answers
390 views

Physical meaning of norm of a matrix

I know norm of a vector is a length of a vector from origin. So what is the motivation behind defining the norm of the matrix? What is the physical meaning of norm of a matrix? Any help is ...
0
votes
1answer
24 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
0
votes
1answer
55 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?
2
votes
1answer
61 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
29 views

Matrix one-norm and infinity-norm

Help me please to find $3\times 3$ matrices $A$ and $B$ under following conditions: $\left \| A \right \|_{\infty }=4\left \| A \right \|_{1}, A \neq 0$ $\left \| B \right \|_{1}=4\left \| B \right ...
1
vote
4answers
60 views

$\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always bounded

Given the matrix $A= (a_{i,j}) \in M_{n,n}$ $||A||=\sup\limits_{x\in X}\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ where $|| $ . $|| _n$ is $ R^N$ norm $\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always ...
0
votes
1answer
14 views

Matrix norm can assume infinite values?

Given a real-valued matrix $A=a_{i,j} \in M_{n,n}$ When: $||A||_2= \sqrt {\sum_{i=0}^n a^2_{i,j}} < + \infty$ ? Why?
1
vote
1answer
32 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 ...
3
votes
2answers
63 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 ...
0
votes
0answers
24 views

Projection and matrix norm

Suppose we are in the matrix space $\mathbf{R}^{n_1 \times n_2}$. Suppose, $R_{\Omega}$ is an operator, such that $R_{\Omega}(Z)$ chooses $m$ entries from $Z$ uniformly at random with replacement and ...
2
votes
1answer
67 views

Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
0
votes
0answers
34 views

norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
2
votes
2answers
53 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
37 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, ...
0
votes
1answer
20 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
2
votes
1answer
47 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 $\arg \min_{X \in \mathbb{R}^{m \times k}} \left|| AX - B\right||_F$, where ...
2
votes
1answer
22 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 $\|A\|_2 \geq \|A'\|_2$?
1
vote
1answer
60 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 ...
0
votes
1answer
33 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
0
votes
1answer
36 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
49 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
37 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
26 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 ...
0
votes
1answer
16 views

derivative of 2 norm wrt matrix

I have a matrix A which is of size m,n, a vector B which of size ...
12
votes
1answer
325 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
47 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
72 views

Showing 2- norm of Ax,squared is equal to $$ x^TM^TMx $$

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 ...
3
votes
1answer
22 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
0
votes
1answer
23 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
2
votes
1answer
71 views

How to FAST calculate 2 norm / spectral norm of a matrix.

I meant reduced 2 norm, the largest singular value. My current approach is applying the SVD decomposition of A via "?gesdd" in MKL, and then taking the largest singular value. I think there should ...
0
votes
0answers
22 views

Matrix - calculate norms

Let be a matrix of $n \times n$ elements. It has on each column $k$ elements equal to $1$ and all the rest equal to $0$. The question is : calculate the minimum and maximum norms $\|A\|_p$ of this ...
2
votes
2answers
40 views

Frobenius norm bound

Is there any way to bound Frobenius norm of a product of square matrices A,B and a vector x in the following way: $$ \|ABx\|≤ \|Ax\|\text{ and }\|B\| $$
1
vote
1answer
29 views

Does this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
1
vote
2answers
72 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
2
votes
2answers
33 views

Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho ...
0
votes
1answer
33 views

Matrix Norm Lemma

There is a lemma claims that : $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|) = ||A|| $ I'd like to know how come $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|)$ because it does not make sense ...
1
vote
1answer
57 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
2
votes
2answers
133 views

Derivative of a matrix: Outer product chain rule

I ran into a seemingly simple matrix calculus question that I can't seem to find the solution to. Suppose I have the following matrices: $X_{(t \times n)}, V_{(n \times m)}$, and $\Phi_{(t\times m)} ...
0
votes
0answers
38 views

l1 minimization with orthogonality constraint

I want to find a rotation (or reflection) for my data which maximizes the space between my points and the basis' margins. I have formulated the problem as follows: Given $X \in \mathbb{R}^{n \times ...
2
votes
2answers
49 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
1
vote
2answers
116 views

Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
0
votes
1answer
30 views

Restrictions on a Matrix-Vector product

Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
1
vote
2answers
44 views

Computing an induced matrix norm

Assume I have a $n \times n$ matrix and a norm defined as $\|A\| = \max \limits_{x \not = 0}\frac{\|Ax\|}{\|x\|}$, where $\|x\| = \sqrt{\sum x_i^2}$. How can I compute this norm?
1
vote
1answer
27 views

Norm of a Matrix-vector product

Suppose I have vector $\vec x \in \mathbb R^n$ and matrix $\mathbf M$ of dimension $m\times n$. Is there an alternative expression for $\lVert \mathbf M \cdot \vec x \lVert$ that includes $\lVert \vec ...
0
votes
2answers
104 views

What is the Hessian of Frobenius norm

As we know that every norm is convex, and if a function is convex w.r.t. the input variable, then corresponding Hessian should be positive semidefinite. When I try to find the Hessian of Frobenius ...
3
votes
1answer
79 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
0
votes
1answer
46 views

Prove relative error with condition number of matrix inequality

I was working on some questions and solutions, and encountered the following question. I am able to prove the inequality using the given information and some algebraic manipulation but the "within ...
0
votes
0answers
18 views

Prove this result about norm 2 of a matrix

$\left | \left | A \right | \right |_{2} :=\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$ Show this: If A is a simetric matrix then $$\left | \left | A ...