Questions tagged [majorization]

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8 votes
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A Muirhead Like Inequality

I am looking for a proof of the inequality as follow: Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ ...
Đào Thanh Oai's user avatar
8 votes
0 answers
374 views

When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?

For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order. ...
Nuno's user avatar
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5 votes
1 answer
722 views

Proving a majorization inequality for the singular value of the product of two matrices without using tensor product

For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds $$ \tag{1} \label{grz} \sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
LayZ's user avatar
  • 115
4 votes
2 answers
417 views

Combination power elementary symmetric polynomial inequality

Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows: Inequality 1: Let $n>2$ and $1 \le m \le n$...
Đào Thanh Oai's user avatar
2 votes
1 answer
531 views

Positive and trace-preserving transformations with a common fixed point of full rank

The following problem which has been on my mind for a while now arises from the realm of quantum information involving quantum channels with a common fixed point of full rank, as well as majorization ...
Frederik vom Ende's user avatar
2 votes
1 answer
247 views

Gale order on multisets of elements of a lattice

The question Let $L$ be a lattice (in the sense of combinatorics, not number theory). An $L$-bag will mean a finite multiset of elements of $L$. Given an $L$-bag $A$, we consider three possible ...
darij grinberg's user avatar
2 votes
1 answer
110 views

Spectral majorization for symmetric matrices

In ${\mathbb R}^n$, a vector $a=(a_1,\ldots,a_n)$ is said to majorize another vector $b=(b_1,\ldots,b_n)$ if for any convex function $f\colon\mathbb R\to\mathbb R$, we have $$\sum_{i=1}^nf(a_i)\ge \...
DRJ's user avatar
  • 170
2 votes
0 answers
344 views

An inequality related to Power sum and elementary symmetric polynomial and majorizes

Power sum and elementary symmetric polynomial Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum: $$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^k =...
Đào Thanh Oai's user avatar
1 vote
1 answer
95 views

A sufficient condition for weak majorization from below

I posed this question on math.stackexchange.com but have gotten no answer. So I post the question here in order to obtain an answer. $\forall x\in \mathbf R^{n+1}$, let $x_{(0)}\le x_{(1)}\le\,\cdots\...
Hans's user avatar
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1 vote
0 answers
219 views

Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?

For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes $x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
Nuno's user avatar
  • 213
1 vote
1 answer
307 views

A sequence and majorization

For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the ...
Toni Mhax's user avatar
  • 640
1 vote
0 answers
43 views

Linear maps that increase majorization order

Let $x$ a vector in $d$ dimensions with positive entries summing to one (a probability distribution). Is there a characterization of the linear operators $T:R^{d}_{+}\to R^{d}_{+}$ such that: $$ x\...
Fabio's user avatar
  • 329
0 votes
1 answer
407 views

Olympiad inequality as a generalizing result due at the origin to Vasile Cirtaoje [closed]

Claim: let $a,b,c>0$ and $p\geq 1$ then we have : $$\left(\frac{a^3}{13a^2+5b^2}\right)^p+\left(\frac{b^3}{13b^2+5c^2}\right)^p+\left(\frac{c^3}{13c^2+5a^2}\right)^p\geq 3\left(\frac{a+b+c}{54}\...
DesmosTutu's user avatar
0 votes
0 answers
36 views

Majorization for vector valued function: looking for literature

Let $x,y\in R^{d}$. A function $f:R^{d}\to R$ is called Schur convex if $$ x\prec y\;\;\rightarrow\;\;f(x)\leq f(y). $$ I am interested in functions $g:R^{d}\to R^{d}$ such that $$ x\prec y\;\;\...
Fabio's user avatar
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