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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).

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||A-B|| compared to the difference of their smallest singular values

I came cross a problem: For two matrices $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{m \times n}$, if ${\rm rank}\ A = {\rm rank}\ B = r \leq \min \{m,n\}$, then \begin{equation} |\sigma_r(A)...
xiuhua's user avatar
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1 answer
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Proof of $\|P\|_2=1$ iff $P$ is an orthogonal projector - continuation

I am trying to do the same exercise as in this question: Let $P\in C^{m×m}$ be a non-zero projector. Show that $||P||_2=1$ iff $P$ is an orthogonal projector. I managed to prove everything but the ...
tigre200's user avatar
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1 answer
42 views

Operator norm with respect to the l-2 norm of the matrix that have identical elements

I am interested in a matrix $A \in \mathbb{R}^{m \times n}, m\geq n$, whose elements are identically $a$. Is there a way to relate $||A||$, the operator norm of the matrix induced by vector 2-norm, to ...
William Lin's user avatar
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Relation between spectral norm and Euclidean norm of a matrix

I'm reading [BGG+'Eurocrypt2014] paper and I doubt that maybe there be a typo in relation between spectral norm and Euclidean norm of a matrix. Here is the part of the paper I think we must have $\|\...
user1035648's user avatar
1 vote
2 answers
71 views

$u,v$ are unit vectors. What is the value of max $|u^TAv|$

Suppose $u,v\in \mathbb{R}^n,A\in\mathbb{R}^{n\times n}$, $u,v$ are unit vectors. The goal is to get the value $\max_{u,v}$ $|u^TAv|$. If $A$ is symmetric, then it can be diagonized. $|u^TAv|= \langle ...
qmww987's user avatar
  • 925
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21 views

Upper bound on the condition number of the similarity transformation matrix

In the context of square matrices, we have been given $$A = T\Lambda T^{-1}$$ where $\Lambda$ is a known diagonal matrix. It is also known that the condition number of A is bounded above. Say, $$\...
Manish Kumar's user avatar
4 votes
1 answer
125 views

Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.

Problem: Show that if $\mathbb{L}(\mathbb{R}^2)$, the space of the linear functionals in $\mathbb{R}^2$, with the matrix norm $\|\cdot\|_p$ ($p > 1$) is isomorphic with the $\mathbb{R}^2$ space ...
Wellington Silva's user avatar
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1 answer
40 views

Expansion of squared L2 norm

Found such expression in a handbook: $$||Ax-y||_2^2=x^TA^TAx-2y^TAx-y^Ty$$ where $A \in R^{n\times n}$ Can't get why this expression holds? I understand that it holds when there is no L2 -norm, as ...
Pyrettt Pyrettt's user avatar
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How much can the operator norm of a matrix increase upon deleting an entry?

By "deleting an entry," I just mean replacing it with $0$. Here's an example. The matrix $\begin{bmatrix} 1&1\\1&-1 \end{bmatrix}$ has norm $\sqrt 2$, but the matrix $\begin{bmatrix}...
Blake's user avatar
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1 answer
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Lower bound of Frobenius inner product

Consider $A,B$ are $n\times n$ matrices. It is known that the Frobenius inner product can be upper bounded as $$\langle A,B \rangle_F \leq \|A\|_2\|B\|_F$$ What about the lower bound?
chloe's user avatar
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operator norm in terms of infinity norm

Given a $n\times n$ matrix $A$ such that $A_{ij}\in[\frac{\sqrt{2}}{2}-1,0]$ What would be the upper bound of operator norm of $A$?
chloe's user avatar
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Induced 2-norm and Frobenius norm inequality

It is well-known that for real matrices $A,B$ of appropriate size, $\| AB \|_F \leq \| A\|_2 \| B\|_F$. What's more, if $B$ is square, we have $\|AB \|_F \leq \| A\|_F \| B \|_2$. However, I want to ...
南洋小學生's user avatar
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gradient of 1-norm of a binary vector

I have some flaw in my logic, which I cannot resolve. It is abouth the gradient of the 1-norm of a binary vector. Let $\mathbf{x} = [x_{1},x_{2},...x_{D}]^{T}$ denote a $D$-dimensional vector and let $...
Dennis Marx's user avatar
2 votes
1 answer
54 views

Matrix norm inequality involving triangle inequality

Consider the following norm: $$ T = ||mge_3 + \frac{m}{k_v}\cdot(k_x\xi + k_1\cdot (\xi-q)+ k_2\cdot(\xi-q+w)|| \ \ (1) $$ where $m,k_v,k_1,k_2 > 0$ constant scalars and $\xi, q, w\in \mathbb{R}^{...
Teo Protoulis's user avatar
1 vote
1 answer
63 views

Show that $\mathcal{L}(\mathbb{R}^2)$ with the norm 2 of matrices is isomorphic to $\mathbb{R}^2$ with norm of vectors $|.|_{2}$

Show that $\mathcal{L}(\mathbb{R}^2)$ with the norm 2 of matrices is isomorphic to $\mathbb{R}^2$ with norm of vectors $|.|_{2}$ I'm struggling with the question itself. What does it even mean? I know ...
Eduardo Alves's user avatar
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14 views

Induced matrix norm, for strange vector norm

I have the following vector norm $||x|| = max[|x_2|,|x_1|+\frac{1}{2} |x_2|], x \in R^2$.How can I compute its induced matrix norm? I have tried several p-norms, but the problem is that I have $|x_1| +...
VadimStacheff's user avatar
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1 answer
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How to split norm square of two matrices' summation

It is known that for any two matrix $A,B$ (not necessarily square), we have $$\forall \alpha >0, \|A+B \|_F^2 \leq (1+\alpha)\|A \|_F^2 +(1+\alpha^{-1}) \| B \|_F^2.$$ This is about the Frobenius ...
南洋小學生's user avatar
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Low rank positive semi-definite approximation of a symmetric matrix

Let $A$ be an $n\times n$ symmetric matrix. Consider the problem $$\min \|A-B\|$$ subject to $B\geq 0$ and $\text{rank}(B)\leq r$, where $\|\cdot\|$ denotes the Frobenius norm. How can I solve this ...
Alphie's user avatar
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2 answers
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p-norm of matrix transpose

I know that $\|A\|_F = \|A^T\|_F$. But I cant find anywhere if $\|A\|_p = \|A^T\|_p$ is true. Is it true? How could I demonstrate it? Thank you!!
ignaciocrb's user avatar
1 vote
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Bound for entry-wise difference of p-unit vectors.

Suppose $p \in [1,2]$ and that $(x,y,z) \in \Bbb{C}^3$ has $p$-norm equal to 1 (that is $|x|^p+|y|^p+|z|^p=1$). When $p=1$ or $p=2$ we get: $$ |x-y|^p+|x-z|^p+|y-z|^p \leq 2^{p-1}+1. $$ This bound is ...
Leo Sera's user avatar
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1 vote
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Norm of the Exponential of a Scalar Multiple of a Matrix

Basically, the title. Suppose, we have some matrix (or operator) $A$ - can we relate $\left\lVert e^{A}\right\rVert_2$ to $\left\lVert e^{k A}\right\rVert_2$, where $k\in\mathbb{R}$? More generally, ...
gettingmathy's user avatar
1 vote
0 answers
12 views

bound for SR1 update

Suppose the exact hessian $H^\star$ as function of vector x (no need to further be specified) and the initial SR-1 approximation $H$ are globally bounded in some norm of your choice by some real ...
user23311233's user avatar
1 vote
0 answers
21 views

Is there a way to efficiently estimate the maximum singular value of a symmetric, non-constant matrix (given some constraints)?

Given a symmetric n x n matrix - whose entries are polynomial functions (let's say of degree 1 but ideally higher degrees as well) of n variables $x_0 ... x_n$ - is there some established method to ...
ufghd34's user avatar
  • 81
1 vote
1 answer
45 views

Invertibility of $A+B$ when $A,B\in\mathbb{R}^{n\times n}$ are symmetric and $A$ is invertible

Here is the full assignment: Let $A,B\in\mathbb{R}^{n\times n}$ be symmetric. In addition, assume that $|\lambda_{amin}(A)|\geq2$ and $|\lambda_{amax}(B)|\leq1$, where $\lambda_{amin}(A)$ is the ...
IMM's user avatar
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Vector $p$-norm submultiplicativity

Let $X=M_n(\mathbb R)$. Is the $p$-norm on $\mathbb{R}^n$ defined by : $$||x||_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$$ with $p \in [1,\infty[$ submultiplicative : $||xy||_p \leq ||x||_p||y||_p$ ...
ztg02's user avatar
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3 votes
4 answers
172 views

$O$ orthogonal with $\det(O)=-1$ implies $||\Omega - O \Omega O^{T}|| = 2 $?

Let $O\in \mathrm{O}(2n)$ be an orthogonal matrix. Let $\Omega$ be the matrix $\Omega:= \bigoplus^n_{i=1} \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}.$ Is it true that: $\det(O)=-1$ ...
Dante Perès 's user avatar
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Schatten p-norm inequality $\|A-B\|_p ≥ \|D_A-D_B\|_p$

I am seeking a reference or proof for the following. Given two Hermitian matrices $A$ and $B$, and their corresponding diagonal matrices $D_A$ and $D_B$ containing their eigenvalues, I am interested ...
Dante Perès 's user avatar
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31 views

Does the accuracy of the iterations of the Newton method transfer to parts of the underlying non-linear equation system?

I'm just wondering one thing. Suppose I have a non-linear system of equations $F(z) = z - d(z) = 0$ for $F: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$. If I apply a Newton method with respect to $...
Donnie's user avatar
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2 votes
1 answer
55 views

Does $x^TAx \approx 1$ imply $||A-xx^T||_2 \approx 0$, where $A$ positive definite and trace 1, $x$ is a unit vector?

Suppose $A$ is a positive definite matrix and $\text{tr}(A)=1$, $x$ is a unit vector. If $x^TAx=1-\delta$ where $\delta>0$ is a small number, can we give a upper bound for $||A-xx^T||_2^2=\text{tr }...
qmww987's user avatar
  • 925
3 votes
1 answer
47 views

Question about the induced two-norm of pseudo-inverse matrix

Given a full row rank matrix $A\in R^{m\times n}$, where $n>m$. Let $A^+$ be the pesudo-inverse matrix of $A$ and $a_{ij}$ be the element of this matrix, where $i\in\{1,\ldots,m\}$ and $j\in\{1,\...
Jeremy's user avatar
  • 69
1 vote
1 answer
74 views

Operator norm of $A^2B$ versus $ABA$

Let $A$ and $B$ be symmetric, positive, invertible matrices. My end goal is to show that $\|A^2B\| = \|ABA\|$ in operator norm. I initially thought it could be proved as follows. By definition of ...
DimSum's user avatar
  • 685
2 votes
0 answers
28 views

Minimize max-norm of 3x3 orthogonal matrix

What is the value of $$ \min_{O\in \mathrm{O}(3)} ||O||_{\max} = \min_{O \in \mathrm{O}(3)} \max_{ij} |o_{ij}|$$ or are there non-trivial bounds known to this quantity? Here, the minimum is over 3x3 ...
Marsl's user avatar
  • 323
1 vote
1 answer
59 views

Maximize Frobenius norm of product under Frobenius norm constraint

Let $\mathbf{A}\in\mathbb{R}^{m\times n}$. I wish to solve $$\mathrm{argmax}_{\mathbf{X}\in\mathbb{R}^{n\times k}} \left\|\mathbf{AX}\right\|_F$$ under constraint $\left\|\mathbf{X}\right\|_F=c$, with ...
Djoudjou's user avatar
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0 answers
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Minimum Hilbert-Schmidt matrix norm with fixed diagonal entries and rank

I want to verify whether the following holds: If $A=(a_{ij})\in M_d(\mathbb{C})$ satisfies $a_{ii}=1$ for $i=1,\ldots, d$ and $\mathrm{rank}(A) = k$, then $\displaystyle \|A\|_2^2=\sum_{i,j=1}^d |a_{...
rpikak's user avatar
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0 votes
1 answer
31 views

How far is the spectral norm from the Frobenius norm?

I know spectral norm is the smaller than any other norm. But can we qualify how small? Formally, for a matrix $A$, what is $||A||_F - ||A||_2$ bounded by? If it is not bounded, can the difference be ...
Debojjal Bagchi's user avatar
-1 votes
1 answer
54 views

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$? [closed]

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$ For example: if $A$ is k$I$ for any scaler $k$, it ...
Debojjal Bagchi's user avatar
1 vote
0 answers
23 views

$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$

I'm currently stuck to the following statement. Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
jason 1's user avatar
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1 vote
0 answers
37 views

Question about Lipschitz continuity of Frobenius norm

Suppose $\sigma: \mathbb R \mapsto \mathbb R^{n\times n}$ is a matrix function and that there exist $\alpha \ge 0$ and $\hat\sigma: \mathbb R \mapsto \mathbb R^{n\times n}$ so that $$\sigma(x)\sigma^T(...
epsilon's user avatar
  • 173
0 votes
0 answers
13 views

Norm inequality for eigendecomposition and oblique projectors

Consider a stochastic matrix R which permits an eigendecomposition into oblique projectors, $$R = \sum_{\lambda} \lambda C_{\lambda}$$ I've observed the following 2-norm inequality in a number of $R$ ...
Renmusxd's user avatar
2 votes
1 answer
30 views

Existence of Limit from Gelfand's Formula

Let $A$ be a square matrix over $\mathbb C$ with spectral radius $\rho(A) > 0$ and $||A||$ denote the operator norm of $A$. By Gelfand's formula, we have $||A^n||^{1/n} \to \rho(A)$ as $n \to \...
Joey's user avatar
  • 113
1 vote
0 answers
19 views

When is $\Vert A+B \Vert = \Vert X + X^\dagger \Vert$ for $[A, X; X^\dagger, B] \succcurlyeq 0$?

Given $$ H = \begin{pmatrix} A & X \\ X^\dagger & B \end{pmatrix} \succcurlyeq 0 , $$ with $A \succcurlyeq 0$ meaning $A$ is positive semi-definite (and Hermitian) and $A, B, X$ are $n\...
Aritra Das's user avatar
  • 3,560
0 votes
0 answers
18 views

Decomposition of a matrix based on vector orthogonality criteria

Given a vector $u\in\mathbb R^{n}$, $u=Wx$, where $x\in\mathbb R^{m}$ and $W$ is a matrix of appropriate dimensions. Let $v\in\mathbb R^{n}$ be a fixed unit vector. I can decompose $u$ into $\perp$ ...
Phoenix's user avatar
  • 103
1 vote
2 answers
56 views

"Best" Submultiplicative / Subordinate norm?

I have $y = Ax$ where x, y are vectors and $A$ is a matrix. I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I ...
Debojjal Bagchi's user avatar
0 votes
1 answer
38 views

Is there any inequality involving the Frobenius norm and the dimension of matrix?

Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following: $$ \left \| AB \right \|_F \leq f(m,r,n)g(A,B) , $$ or $$ \left \| AB \...
ai zhongguo's user avatar
1 vote
0 answers
47 views

Pythagorean theorem for infinity norm [closed]

I am trying to prove or provide a counter-example for the following statement: If $x$ and $y$ are two complex vectors in $\mathbb{C}^n$ that are perpendicular to each other, i.e., $x^{H} y = 0$ where $...
Poorya Mollahosseini's user avatar
0 votes
1 answer
48 views

Upper bounds for $\| I - \boldsymbol{1}\boldsymbol{p}^T \|_2$

I want to know if we can have some bound of $\| I - \boldsymbol{1}\boldsymbol{p}^T \|_2$, where $I$ denotes the identity matrix of size $k$, and $\boldsymbol{p} \in \mathbb{R}^K$ is some known vector ...
南洋小學生's user avatar
1 vote
0 answers
45 views

Norm Inequalities in Matrix Multiplication

Consider $B\in \mathbb{R}^{m\times h}$ and $A \in \mathbb{R}^{h\times n}$. We can make sense of \begin{align*} \lVert BA\rVert_{1,\infty} = \sup_{\{x \in \mathbb{R}^n:\lVert x\rVert_1 = 1\}}\lVert ...
yf297's user avatar
  • 79
0 votes
1 answer
70 views

Is infinity norm submultiplicative? What about its power?

Here is the setting $X$: Positive semi-definite; Non-negative entries; $||X||_\infty = \lambda$ (largest absolute row sum is $\lambda$). $G$: Diagonal matrix; Non-negative diagonal entries $g_i$; All $...
Xu Siyuan's user avatar
0 votes
2 answers
164 views

Multidimensional Mean Value Theorem with arbitrary norm

In the question Multivariate Mean Value Theorem Reference was written the following statement for $x,y\in \mathbb{R}^{n}$ \begin{equation} ||f(x) - f(y)||_q \leq \sup_{z\in[x,y]}||f'(z)||_{(q,p)}||x-...
Иван Петров's user avatar
0 votes
0 answers
18 views

Proof of an inequality regarding the norm of a matrix polynomial

I've found this inequality being used a lot in papers but have been unable to prove it for myself. Consider a matrix polynomial $P(z)=A_{m}z^{m}+...+A_0$ with complex matrix coefficients. Then, for ...
Eleanore's user avatar

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