Questions tagged [matrix-norms]
This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
686
questions
0
votes
0
answers
30
views
Prove an equation about Frobenius norm
The problem:
Show that if $\textbf{0} \neq \textbf{v} \in \mathbb{R}^{n}$ and $E \in \mathbb{R}^{n\times n}$, then
$$\Big\lVert E(I - \frac{\textbf{v}\textbf{v}^{T}}{\textbf{v}^{T}\textbf{v}})\Big\...
- 13
1
vote
1
answer
80
views
Prove $\lVert A \rVert _{p} \geq 1$ for any idempotent matrix $A \neq 0$
As the title states,
I'm facing a problem to prove:
for any idempotent matrix $A \in \mathbb{C}^{n \times n}$ and $A \neq 0$, $\lVert A \rVert_{p} \geq 1$.
Here the p-norm $\lVert A \rVert_{p}$ ...
- 13
0
votes
0
answers
26
views
Triangle inequality for weighted norms
I am trying to prove or disprove the triangle inequality for the function:
$f(X) = \sqrt{\sum\limits_{k=1}^{d}\frac{|X_{k}|^{2}}{k}}$ ; where $X \in R^{d}$.
I tried whatever I could, and my ...
- 135
1
vote
1
answer
49
views
Monotonicity of Frobenius norm
$A,B\in \mathbb{R}^{n\times n}$,$A,B\succeq 0$,$B-A\succeq 0$, prove $\Vert A \Vert_F \leq \Vert B \Vert_F $
(If $X \succeq 0$, then there exists
another matrix $Y \succeq 0$ such that $Y^2 = X$. ...
- 11
0
votes
0
answers
18
views
F-norm relation to singular value
$\|A\|_F = \sqrt{tr(A^*A)} = \sqrt{tr(AA*)} = \sum_k \sigma_k$
Is the last equality true in general?
user1220545
0
votes
0
answers
12
views
Relation between singular value, eigen value and 2-norm
I am confused.
We have that,
$ \sigma_{max} \ge \rho(A) = \|A\|_2$.
where $ \rho(A) = |\lambda_{max}|$ and $\sigma$ is the singular value of A.
And, $\sigma^2 = \lambda(A^*A)$.
But, $\rho(A) \ne \sqrt{...
user1220545
0
votes
0
answers
14
views
Norm of a Matrix Vector Product : Inequality [duplicate]
I'm trying to look for a concise proof for the inequality ||M.v|| ≤ ||M||.||v|| where v is a vector in R^n and M is a matrix with dimensions m*n.
I get the intuition behind what the inequality is ...
- 11
0
votes
1
answer
21
views
$\Sigma$ norm in Normal Distribution
My professor wrote down this in class:
Let $D=N(\mu,\Sigma)$ be a normal distribution, then $$\log(p_D(d))=-\frac{1}{2}||d-\mu||^2_\Sigma+\text{const.}$$
Here I don't completely understand from where ...
- 158
2
votes
0
answers
42
views
Estimates of a matrix norm in general relativity
Let $\bar{D}_{2\rho}$ be a closed disk in $\mathbb{R}^2$ of radius $2\rho$, where $\rho>1$.
Let $\psi_0$ be a smooth map：
$$\psi_{0}:[0,1]\times\bar{D}_{2\rho}\rightarrow\hat{S},$$
where $\hat{S}$ ...
- 71
0
votes
2
answers
36
views
2-norm of transpose proof
I don't understand the proof of ‖x‖2=‖xT‖2.
I know you can use the Cauchy-Schwartz inequality and that you can prove it by first showing ‖xT‖2 ⩽ ‖x‖2 and then ‖x‖2 ⩽ ‖xT‖2. I get proving both ...
- 11
1
vote
1
answer
23
views
Relation between 2-norm and F-norm
$\|A\|_2 = \sup_{\|x\|_2 = 1} \|Ax\|_2 = \sup_{\|x\|_2 = 1} \|Ax\|_F$
since A is a matrix and x is a vector, then $Ax$ is a vector.
And we have that $\|x\|_2 = \|x\|_F$, right?
user1220545
0
votes
1
answer
38
views
Matrix norm identity. Understanding inequallity.
Let A $\in R^{m\times n}$. What dose it mean when the book then write A : $R^{n} \rightarrow R^{m}$?
I try to understand this inequality:
$\frac{1}{\sqrt m} \|\mathbf{A}\|_1 \le \|\mathbf{A}\|_2 \le \...
user1220545
0
votes
1
answer
36
views
L2 norm of vector samples drawn from uniform distribution [-1,1] and standard normal distribution
I was given a problem to make a guess about the expected values of "l2-norm square" of random vectors of size n which are drawn from a uniform distribution of [-1,1] and standard normal ...
0
votes
0
answers
33
views
Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what we can say about Frobenius ...
- 53
0
votes
1
answer
59
views
Explanation of norm of cross product formula
I have a line between two points given as $(x_1, y_1)$ and $(x_2, y_2)$ in Python code:
...
- 135
1
vote
0
answers
39
views
Nuclear norm of matrix comprised of points sampled uniformly at random from the hypersphere?
In a subfield of machine learning called "self-supervised learning", many methods constrain network representations to lie on the hypersphere. I want to understand how such representations ...
- 453
1
vote
1
answer
31
views
Show the Schatten norm inequality $\left\| \boldsymbol{v} \right\| _{2/3}^{2}\ge \left\| \boldsymbol{v} \right\| _{1}^{3}$?
I want to show the following inequality
$$
\left\| \boldsymbol{v} \right\| _{2/3}^{2}\ge \left\| \boldsymbol{v} \right\| _{1}^{3} \tag 1
$$
where the norm is defined as $\left\| \boldsymbol{v} \right\|...
- 87
3
votes
0
answers
61
views
Proof Map is (Inf) Norm ||A||$_∞$
so I have learned to proof the 1-Norm,2-Norm and p-Norm.
I know that max is a function, which returns the maximum in a list/array.
So in this example I have:
$\|A\|_\infty$
$$
\|A\|_\infty := \max_{1\...
- 31
0
votes
1
answer
69
views
A Property of the Trace-Norm
Let, $\mathcal{E}$ be a Completely-Positive Trace-Preserving Map, i.e, for linear operators $\rho$ on a Hilbert-Space,
$$
\mathcal{E}(\rho) = \sum_i E_i\rho E_i^{\dagger}\qquad {\rm s.t.}\qquad \sum_i ...
0
votes
1
answer
22
views
Error on trace of quadratic forms from Frobenius error bound on central matrix
$\DeclareMathOperator{\Tr}{Tr}$
If a Frobenius error on an estimate of covariance $\|\tilde{\boldsymbol{M}}-\boldsymbol{M}\|_F$ is known to be in the order of $\tilde{O}(f(n,d))$ as some function of $...
- 191
1
vote
0
answers
25
views
Interpretation/terminology of a matrix
I have two vectors ${\bf x} \in \mathbb{R}^d$ and ${\bf y} \in \mathbb{R}^p$ such that $d < p$ and let ${\bf A} \in \mathbb{R}^{s\times d}$ and ${\bf B} \in \mathbb{R}^{s\times p}$two matrices, ...
- 105
0
votes
1
answer
31
views
Norm or metric that takes into account the proximity of entries of two matrices
I was wondering whether there exists a crisp norm or metric on $\mathbb R^{n \times n}$ that has the following behaviour:
Let $A, B \in \mathbb R^{n \times n}$ two matrices. Now I would like them to ...
- 23
0
votes
1
answer
19
views
Please help with proving or disproving the statement related to vector induced norm of real symmetric matrices.
Consider any vector induced norm (need not be just p-norm). I need to prove or disprove that for any real symmetric matrix A and vector induced norm $||.||$,
$$
||A|| = \rho (A)
$$
where $\rho(A)$ is ...
- 13
0
votes
0
answers
96
views
Showing the upper bound of the operator norm of a rectangular matrix
Question: Given a square random matrix $X\in\mathbb{R}^{p\times p}$ where
$$
X_{ij}\stackrel{iid}{\sim}\text{Bernoulli}(\alpha),
$$
where $\alpha\in(0,1)$, satisfies the condition
$$
\|X\|_{\text{op}}&...
- 664
0
votes
1
answer
29
views
Bounding the growth/decay of a linear ODE system
Suppose we have a linear system of differential equations governed by a matrix $A$:
$$ \frac{d\mathbf{x}}{dt} = A\mathbf{x}, \ \mathbf{x}(t=0) = \mathbf{x}_0$$
We know that the solution is
$$ \mathbf{...
- 43
2
votes
0
answers
90
views
On the matrix norm inequality $\|A B\| \ge \|A\|\cdot \|B\|$
I know that $\|A B\| \le \|A\|\cdot \|B\|$. Are there any inequalities like the form: $\|A B\| \ge c \,\|A\|\cdot \|B\|$?
- 21
0
votes
0
answers
31
views
Upper bounding $\|\sum_{k = 0}^{N} \frac{(At)^k}{k!}\|$
I'm trying to figure out the tightest upper bound for
$$\left \|\sum_{k = 0}^{N} \frac{(At)^k}{k!} \right\|,$$
where $A$ is a matrix and $\|A\|t > 1$. I know that $e^{\|A\|t}$ is a known upper ...
- 43
0
votes
0
answers
77
views
More general traceless normalized matrix
I am wondering about how to characterise the more general traceless matrix over $\Bbb C$
$$ \operatorname{Tr} (T) = 0$$
and having unit Hilbert-Schmidt norm,
$$ \| T \|^2 = \operatorname{Tr} \left( T ...
- 91
1
vote
0
answers
54
views
Upper bound of biggest singular value of Kronecker Sum via singular values and traces
I am investigating the biggest singular value of Kronecker Sum. I'd like to understand if my argument is correct.
Let $A$, $B$ be square matrices of size $n$. Let $I$ be an identity matrix of size $n$....
0
votes
0
answers
27
views
Computing the inf-norm of the inverse of a matrix
Consider a regular matrix $A \in \mathbb{R}^{n \times n}$. The normal $\infty$-norm of $A$ is given by
$$
\|A\|_{\infty} = \max_{\|x\|_{\infty} = 1} \|Ax\|_{\infty}
= \max_{i \in [n]} \sum_{j \in [n]}...
- 161
1
vote
0
answers
25
views
Applying Matrix 2-Norm in Proof - Is this reasoning incorrect?
Thank you in advance for helping out an apsiring mathematician! :)
I am currently self-studying matrix norms (in the context of machine learning but that is not necessary for my question), and I came ...
- 11
0
votes
1
answer
41
views
Norm of sum of positive matrices versus the sum of norms
Suppose that $P_1, \dots, P_k$ are positive $n \times n$ matrices.
Is it true that
$$ \sum_{k} \| { P_k } \| \leq n \left\| {\sum_k P_k} \right\|,$$
where $\| \cdot \|$ denotes the operator norm?
My ...
1
vote
0
answers
67
views
How to minimize $\| x {\bf I} - {\bf A} \|_2$?
Given the matrix ${\bf A} \in {\Bbb R}^{n \times n}$,
$$ \begin{array}{ll} \underset {x \in {\Bbb R}} {\text{minimize}} & \left\| x {\bf I}_n - {\bf A} \right\|_2 \end{array} $$
where $\| \cdot \|...
0
votes
2
answers
62
views
Prove that there is a rank one matrix $B$ so that $Bx = y$ where $B$ has matrix norm $1$
Let $\lVert \cdot \rVert$ denote a norm on $\mathbb{C}^m$. Define the dual norm $\lVert \cdot \rVert '$ by $\lVert x\rVert' := \sup_{\lVert y\rVert = 1} |y^* x|,$ where $y^*$ is the conjugate ...
- 153
2
votes
1
answer
81
views
What is the "matrix-norm" of a vector?
What does this notation mean?
$$
||\mathbf{x}||_{A}
$$
$\mathbf{x} \in \mathbb{R}^n$
$A \in \mathbb{R}^{n \times n}$ a square matrix (positive definite, if that happens to be relevant).
Searches for ...
- 23
2
votes
1
answer
87
views
If $L$ is a positive semi-definite matrix and all elements in each row add up to 0, How to prove that the inf-norm of $(I+L)(I+L+L^2)^{-1} \leq 1$?
Assuming that the L is a positive semi-definite matrix and
$$ \sum\limits_{j=1}^{N} L_{i,j} = 0. \quad for \; i = 1,...,N.$$
I want to prove that $\|(I+L) (I+L+L^2)^{-1}\|_{\infty} \leq 1$, I have ...
- 21
2
votes
1
answer
56
views
Prove vector norm inequalities and use the Schwarz Inequality to confirm ratio bound
Two Part Problem: Prove that $||x||_\infty \leq ||x|| \leq ||x||_1.$ Show from the Schwarz Inequality that the ratios $||x|| / ||x||_\infty$ and $||x||_1 / ||x||$ are never larger than $\sqrt n$. ...
- 101
1
vote
1
answer
64
views
For a symmetric matrix $A$, $\|A\|_2 \leq \|A\|_{\infty}$? [duplicate]
I am reading the following paper.
Yuval Dagan, Constantinos Daskalakis, Nishanth Dikkala, Surbhi Goel, Anthimos Vardis Kandiros, Statistical Estimation from Dependent Data, ICML 2021.
From page 6 ...
- 171
2
votes
1
answer
73
views
Proof of the variational formulation of the nuclear norm [closed]
How to show that the nuclear norm can be written in the following way?
$$\|X\|_* = \min\limits_{A,B: AB=X}\frac{\|A\|^2_2}{2} + \frac{\|B\|^2_2}{2}$$
Related: Variational characterization of nuclear ...
- 7,299
0
votes
1
answer
68
views
Properties of SPD Matrix Norms
Question:
Let $A,B\in\mathbb{R}^{n\times n}$ be two symmetric and positive definite matrices with $AB=BA$ and $\left(A\underline{x},\underline{x}\right)\geq\left(B\underline{x},\underline{x}\right)\,\...
0
votes
1
answer
81
views
Maximal spectral norm of balanced $\pm 1$ matrix [closed]
Suppose that we have a square matrix $A = [a_{ij}] \in {\Bbb R}^{n \times n}$ whose entries are $\pm 1$ and whose columns are balanced, i.e., $\sum_{i=1}^n a_{ij}=0$. How large can the spectral norm ...
- 381
1
vote
2
answers
64
views
Proving that the norm $n \|A\|_{l_\infty}$ is a matrix norm.
I'm trying to prove that the norm $\| \circ \| = n \|A\|_{l_\infty}$ is a matrix norm. Reminder that the $l_\infty$-norm of a matrix $A \in \mathbb{R}^{n \times n }$ is defined as:
\begin{equation}
\|...
- 79
0
votes
1
answer
101
views
Need a way to find $a,b$ for the vector $x' = ax + by$ such that for a matrix $A$ of size $ m \times n$, $Ax'$ is orthogonal to $Ay$
Let $m, n \in \mathbb{N}^+$ such that $$1 \leq m \leq \frac12 n < n \leq 100$$ denote the known numbers of rows and columns of an unknown matrix $A \in \Bbb R^{m \times n}$. Though one does not ...
- 1
0
votes
1
answer
76
views
If $A$ is a positive semi-definite matrix, is $\left\|I-\frac{A}{\|A\|}\right\|\leq 1$?
I was reading this Wikipedia page and found the following confusing:
In that case($A$ begin positive semi-definite), we have
$$\left\|I-\frac{A}{\|A\|}\right\|\leq 1$$
It is not clear to me what ...
- 5,630
0
votes
1
answer
85
views
Norm ratio: find the impact of diagonal pertubations
Given the matrix
$$P(D)=I-A(A^HA+D)^{-1}A^H$$
that is parameterized on the diagonal matrix $D$ whose diagonal entries are not necessarily the same, and $A$ is full column rank, I want find the impact ...
- 389
-1
votes
1
answer
35
views
Matrix norm definition with constraint
I know that the p-norm of a matrix is defined as:
$\displaystyle \|A\|_p = \max_{x\neq0}\frac{\|Ax\|_p}{\|x\|_p}$
However, how can I conclude the below formula from the above formula?
$\|A\|_p=\max_{\|...
1
vote
0
answers
42
views
Bound on minimum perturbation of eigenvalues, based on condition number
Problem: This problem comes from a past Ph.D. qualifying exam at my institution.
Let $\newcommand{\R}{\mathbb{R}}
\newcommand{\l}{\lambda}
A \in \R^{n \times n}$ be of full rank and diagonalizable as ...
- 41.6k
1
vote
1
answer
50
views
Relation between matrix norms and contractions
I'm not so familiar with matrix norms, so I attempted an elementary proof to find conditions on the map $\phi(x) = Ax + b$ to be a contraction under the metrics induced by the $L^1$ and $L^2$ norms. ...
- 319
1
vote
0
answers
87
views
Prove that $\sum_{k=1}^n|\lambda_k(X)-\lambda_k(Y)|\leq\|X-Y\|_1$
$X,Y$ are $n\times n$ Hermitian matrices. $\lambda_k(X)$ denotes the $k$th largest eigenvalue of $X$. Prove that $\sum_{k=1}^n|\lambda_k(X)-\lambda_k(Y)|\leq\|X-Y\|_1$. (Here $\|X-Y\|_1=\sum_{k=1}^n|\...
- 797
0
votes
0
answers
25
views
Does a bound on the operator norm of a square matrix generalize to a rectangular matrix?
Setup: Given that I have a random square matrix $X$ where each entry is generated independently and identically. Denote $\|X\|_{\text{op}}$ to be the operator norm of the square matrix.
Question: ...
- 664