All Questions
11 questions
5
votes
1
answer
197
views
The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix
Given any $n\in\mathbb N$, consider the the Sylvester-Hadamard-Walsh matrix $M=(a_{i,j})_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let
$$\nu_{n,p}=\max_{F\subseteq 2^n}\...
- 128k
2
votes
1
answer
337
views
Operator norm of triangular truncation on symmetric matrices
Inspired by this question.
It is known that for the matrix $T_n \in \mathcal{M}_n$ (the space of real-valued $n \times n$ matrices) defined by
\begin{equation*}
(T_n)_{ij} = \begin{cases}
1 & i \...
- 83
8
votes
1
answer
678
views
Inequality involving tensor product of orthonormal unit vectors
Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
- 114k
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align*}
\|v^2\| \, \|w^2\| - \langle ...
- 1,991
21
votes
0
answers
868
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 13.4k
2
votes
0
answers
147
views
Is the following inequality true for the norm of Moore-Penrose pseudoinverses?
Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
- 1,465
13
votes
2
answers
1k
views
A matrix norm inequality
Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
- 1,748
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
1
vote
0
answers
85
views
What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
3
votes
2
answers
171
views
a monotone relation for s-numbers
Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one $s_n(...
- 28.6k
2
votes
1
answer
942
views
A singular value inequality
Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
$$\left|s_{1}\...
- 28.6k