Questions tagged [majorization]
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14
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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
$$\sum_{i=1}^nf(a_i)\ge \...
- 170
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When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?
For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order.
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Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?
For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes
$x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
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A sufficient condition for weak majorization from below
I posed this question on math.stackexchange.com but have gotten no answer. So I post the question here in order to obtain an answer.
$\forall x\in \mathbf R^{n+1}$, let $x_{(0)}\le x_{(1)}\le\,\cdots\...
- 2,169
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A sequence and majorization
For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the ...
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Proving a majorization inequality for the singular value of the product of two matrices without using tensor product
For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
- 125
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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 :
$$\left(\frac{a^3}{13a^2+5b^2}\right)^p+\left(\frac{b^3}{13b^2+5c^2}\right)^p+\left(\frac{c^3}{13c^2+5a^2}\right)^p\geq 3\left(\frac{a+b+c}{54}\...
- 107
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Linear maps that increase majorization order
Let $x$ a vector in $d$ dimensions with positive entries summing to one (a probability distribution). Is there a characterization of the linear operators $T:R^{d}_{+}\to R^{d}_{+}$ such that:
$$
x\...
- 329
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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
$$
x\prec y\;\;\...
- 329
2
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1
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470
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Positive and trace-preserving transformations with a common fixed point of full rank
The following problem which has been on my mind for a while now arises from the realm of quantum information involving quantum channels with a common fixed point of full rank, as well as majorization ...
2
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1
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Gale order on multisets of elements of a lattice
The question
Let $L$ be a lattice (in the sense of combinatorics, not number theory).
An $L$-bag will mean a finite multiset of elements of $L$.
Given an $L$-bag $A$, we consider three possible ...
- 32.2k
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Combination power elementary symmetric polynomial inequality
Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows:
Inequality 1: Let $n>2$ and $1 \le m \le n$...
- 1,014
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A Muirhead Like Inequality
I am looking for a proof of the inequality as follow:
Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ ...
- 1,014
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An inequality related to Power sum and elementary symmetric polynomial and majorizes
Power sum and elementary symmetric polynomial
Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum:
$$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^k =...
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