Questions tagged [majorization]
The majorization tag has no summary.
20 questions
3
votes
1
answer
113
views
Stochastic order of generalized chi-square distributions (Extended)
I seek a generalization of the idea and answer found in
Stochastic order of generalized chi-square distributions.
Given that $χ^2_a=∑^n_{i=1}a_iZ^2_i$ where $Z_i∼N(0,1)$ are i.i.d. $\forall i$. The ...
5
votes
1
answer
281
views
Stochastic order of generalized chi-square distributions
For any $a\in \mathbb{R}^n_{>0}$ define the random variable $\chi^2_a=\sum_{i=1}^n a_i Z_i^2
$
where the $Z_i\sim \mathcal{N}(0,1)$ are i.i.d. Given $a,b\in \mathbb{R}^n_{>0}$ I am looking for ...
- 105
2
votes
1
answer
87
views
Spectraly extremal connected trees, a conjecture on majorisation of Laplacian eigenvalues
The precise conjecture that I want to prove or disprove is the following:
Let $P_n$ denote the path graph with n vertices, $S_n$ the star graph with n vertices, and $T_n$ an arbitrary connected tree ...
3
votes
1
answer
245
views
Stochastic Dominance for Beta Distributed random variables
Is it possible to prove that a $Beta(\alpha,\beta)$ distributed random variable is always stochastically dominated by a $Beta(\alpha',\beta')$ if $\alpha'>\alpha$ and $\beta'<\beta$? I've read ...
2
votes
0
answers
157
views
Majorization theory on $\sigma$-finite measure spaces
I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of a finite measure spaces but I'm ...
- 809
0
votes
1
answer
125
views
On a differential inequality with an additional constraint
I am stuck on this problem from a research question, which seems to require solving a differential equation, but I am not sure how to deal with integrals like $\int_0^t$ or $\int_t^1$. I will be ...
- 33
2
votes
1
answer
215
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
$$\sum_{i=1}^nf(a_i)\ge \...
- 286
8
votes
0
answers
471
views
When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?
For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order.
...
- 269
1
vote
0
answers
324
views
Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?
For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes
$x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
- 269
1
vote
1
answer
156
views
A sufficient condition for weak majorization from below
I posed this question on math.stackexchange.com but have gotten no answer. So I post the question here in order to obtain an answer.
$\forall x\in \mathbf R^{n+1}$, let $x_{(0)}\le x_{(1)}\le\,\cdots\...
- 2,363
2
votes
2
answers
504
views
A sequence and majorization
For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the ...
- 907
5
votes
1
answer
1k
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Proving a majorization inequality for the singular value of the product of two matrices without using tensor product
For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
- 115
-1
votes
1
answer
541
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 :
$$\left(\frac{a^3}{13a^2+5b^2}\right)^p+\left(\frac{b^3}{13b^2+5c^2}\right)^p+\left(\frac{c^3}{13c^2+5a^2}\right)^p\geq 3\left(\frac{a+b+c}{54}\...
1
vote
0
answers
66
views
Linear maps that increase majorization order
Let $x$ a vector in $d$ dimensions with positive entries summing to one (a probability distribution). Is there a characterization of the linear operators $T:R^{d}_{+}\to R^{d}_{+}$ such that:
$$
x\...
- 359
1
vote
1
answer
97
views
Majorization for vector valued function: looking for literature
Let $x,y\in\mathbf R^{d}$. A function $f:\mathbf R^{d}\to\mathbf R$ is called Schur convex if
$$
x\prec y\;\;\rightarrow\;\;f(x)\leq f(y).
$$
where $\prec$ is the majorization order.
I am interested ...
- 359